int x/sqrt(x^4-6x^2+5) dx Use integration tables to find the indefinite integral.

Recall that indefinite integral follows int f(x) dx = F(x) +C where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as the constant of integration
The given integral problem: int x/(x^4-6x^2+5) dx resembles one of the formulas from the integration table. We follow the integral formula for rational function with roots as:
int (dx)/sqrt(ax^2+bx+c) = 1/sqrt(a)ln|2ax+b+2sqrt(a(ax^2+bx+c))| +C .
For easier comparison, we apply u-substitution by letting: u=x^2
then du= 2x dx or (du)/2 =xdx .
Plug-in the values, we get:
int x/(x^4-6x^2+5) dx =int 1/(x^4-6x^2+5)*x dx
                                 =int 1/(u^2-6u+5)*(du)/2
Apply the basic integration property: int c*f(x) dx = c int f(x) dx .
int 1/(u^2-6u+5)*(du)/2 = 1/2int 1/(u^2-6u+5) du
By comparing ax^2+bx+c  with u^2-6u+5 , we determine the corresponding values as: a=1 , b=-6 ,and c=5 .
Applying the aforementioned formula for rational function with roots, we get:
1/2int 1/(u^2-6u+5) du
=1/2 * [1/sqrt(1)ln|2(1)u+(-6)+2sqrt(1(1u^2+(-6)u+5))|] +C
=1/2 * [1/1ln|2u-6+2sqrt(u^2-6u+5)|] +C
=(ln|2u-6+2sqrt(u^2-6u+5)|)/2 +C
Plug-in u = x^2  and u^2=x^4   on   (ln|2u-6+2sqrt(u^2-6u+5)|)/2 +C , we get the indefinite integral as:
int x/(x^4-6x^2+5) dx =(ln|2x^2-6+2sqrt(x^4-6x^2+5)|)/2 +C
           

What is the significance of gender in Othello?

Gender is extremely significant in Othello. Iago uses gender norms common in Venice at that time to manipulate Othello into killing the woman he loves.
The stage for this is set early in the play, when Iago hears Brabantio, Desdemona's father say to Othello in act I, scene 3:

Look to her, Moor, if thou hast eyes to see: / She has deceived her father, and may thee.

At this time, gender norms strictly dictated that a woman must be completely sexually faithful to her husband. But, at the same time, many men believed a prejudiced gender stereotype that said women were inherently unfaithful, always ready to have affairs with other men. This is implied in Brabantio's words, the words of a sober patriarch: women are untrustworthy. Iago himself seems to believe this as part of his generally dark view of human nature. He plays on it ruthlessly to plant doubts in Othello's mind about Desdemona.
Because of the honor code that said one's manhood was undermined if one's wife was unfaithful, and because it was considered acceptable to kill an unfaithful wife, Othello kills his innocent spouse. If gender rules had not been so strict, the play could have had a more reasonable outcome.

What does Primo Levi means when he says, "It was my good fortunate to be deported to Auschwitz in only 1944"?

Auschwitz was in operation during the years of 1940–1945, and, as the war progressed in Europe, the Nazi regime changed, and their treatment of prisoners changed. For starters, as the camp closed down and released its prisoners in the middle of 1945, being sent there in 1944 meant you had a year or possibly less to stay in Auschwitz. Obviously this is much better than being sent there in 1940 and trying to survive for 5 or more years in its cruel treatment and risking being sent to the gas chamber.
Additionally, as the war went along, soldiers were drawn away from the camps to active duty more often, so the number of guards to observe and punish prisoners became depleted, and the camp became less organized as time went on. Because of this, while it was still a tormenting place, prisoners were punished slightly less often and less severely and in many cases were less likely to be sent to the gas chamber than earlier on in the war.

As Levi goes on to explain after making the statement about his good fortune, he was lucky to arrive at Auschwitz in 1944 and not earlier because, by 1944, the Germans had decided to treat the Auschwitz prisoners slightly more humanely than they had before.
It was late in World War II, and Germany was losing—with heavy casualties. Therefore, the country was short on the manpower in the concentration camps that helped supply the war machine. The Germans decided to improve conditions so that prisoners would survive longer, in order to get more work out of them before they were eventually killed. For example, food rations were increased so that the prisoners would not starve so quickly. Further, guards were restrained from killing individual prisoners. These factors increased Levi's chances for survival in a brutal environment.
Nevertheless, the conditions Levi describes are horrible, so to some extent Levi's statement is ironic. Of course, it was anything but "good fortune" to end up in a Nazi concentration camp; the fact that conditions were not as absolutely horrible as they could have been was just the slimmest of good luck in a terrible situation.

What is the irony in "The Story of an Hour"?

Irony is created when there is a discrepancy between what we expect to happen and what actually happens. So, one major irony of this story is created when Louise Mallard reacts with joy to news of her loving husband's death. At first, she bursts into tears "with sudden, wild abandonment" and a "storm of grief." It is, therefore, even more of a surprise when Louise goes alone to her room and says to herself, "free, free, free!" So, she is happy about the loss of this husband who, apparently, "never looked save with love upon her"?! The simple answer is yes. She is not rejoicing in his death necessarily but in her own newfound freedom, but she is rejoicing, and this unexpected response to such seemingly tragic news is ironic.
Louise's sister, Josephine, kneels outside her door, concerned for Louise's health, as she fears that her sister is overwhelmed by grief. We, of course, know this to be untrue: if Louise is overwhelmed by anything, it is her happiness that "There would be no one to live for her during those coming years; she would live for herself." Because we, the readers, know more than Josephine, a character, dramatic irony is created.
Again, watching Louise descend the steps with "a feverish triumph in her eyes" as she walks "like a goddess of Victory" adds to the earlier irony of her initial reaction, because we would not expect a woman whose loving husband has just been killed to feel triumphant or victorious. It is also ironic that Brently Mallard walks through his front door as though nothing has happened because, all along, we expect him to be dead (since his friend verified the news by a "second telegram" before coming to break it to Louise). We, and they, expect him to be dead, not unlocking his front door.
Finally, the cause of Louise's death as proclaimed by the doctors, that she died of "joy that kills," is another example of dramatic irony. We know that her joy didn't kill her; instead, it was likely her disappointment that all the freedom she thought would be hers was suddenly snatched away when she realized her husband was still alive.

How is symbolism used in "The Lady with the Pet Dog" to portray the theme of isolation?

The theme of isolation in “The Lady with the Pet Dog” is portrayed through darkness and silence that often obscures the characters from each other and through lonely images associated with travel. On the night that Anna and Dmitri first sleep together, they go to the harbor to watch a ship come in. The ship takes a long time to arrive, and by the time it docks it is “too dark to see people’s faces” (5). The wind dies away and the people from the ship depart so that it is still, silent, and pitch black when he suggests they go back to Anna’s room.
Later, when Anna goes home to take care of her sick husband, Dimitri stands on the train platform: “The train moved off rapidly, its lights soon vanished from sight, and a minute later there was no sound of it, as though everything had conspired together to end as quickly as possible that sweet delirium, that madness.” Again, there is a lone transport vehicle, silence, and darkness, separating the characters.
Even in the final scene when it seems they may finally find a way to be together, the blocking creates a sense of separation. Anna keeps her back to Dmitri so that he can’t see her face. When he goes to her at the window, he looks not at her, but at himself. He notices that he looks old and realizes that women never see him as he sees himself: “He always seemed to women different from what he was, and they loved in him not himself, but the man created by their imagination” (19).
http://www.shortstoryamerica.com/pdf_classics/chekhov_lady_withthe_pet_dog.pdf

10^(3x)+4=9 Solve the equation.

For the given equation 10^(3x)+4 =9 , we may simplify by combining like terms.
Subtract 4 from both sides of the equation.
10^(3x)+4-4 =9-4
10^(3x)=5
Take the "ln" on both sides to be able to bring down the exponent value.
Apply the natural logarithm property: ln(x^n)= n*ln(x) .
ln(10^(3x))=ln(5)
3xln(10)=ln(5)
To isolate the x , divide both sides by 3ln(10).
(3xln(10))/(3ln(10))=(ln(5))/(3ln(10))
x=(ln(5))/(3ln(10))
x= (ln(5))/(ln(1000)) or 0.233 (approximated value).
Checking: Plug-in x=0.233 on 10^(3x)+4 =9.
10^(3*0.233)+4 =?9
10^(0.699)+4 =?9
5.00034535+4=?9
9.00034565~~9   TRUE.
Therefore, there is no extraneous solution.
The x=(ln(5))/(3ln(10)) is the real exact solution of the given equation 10^(3x)+4 =9 .

What is Montag's reaction when he discovers his wife has taken an overdose of sleeping pills?

In Part One, Montag returns home late from work and accidentally kicks an empty pill bottle that is lying on the floor of his house. After entering his bedroom and seeing that the bottle of sleeping pills is empty, he discovers his wife lying motionless on their bed. Montag reacts by yelling Mildred's name and is immediately overwhelmed by the sound of jets flying over his house. The jets flying overhead shake Montag's home and consume his attention as he screams at the top of his lungs. After the jets pass, Montag calls emergency responders to revive his wife, who has overdosed on sleeping pills. Shortly after calling the emergency responders, two medical operators arrive at his home with a snake-like device that flushes Mildred's system and replaces her blood. The entire experience is unnerving for Montag, who is not only concerned about his wife's health but also her mental state. Overall, Montag is initially frightened and worried when he discovers that his wife has consumed all of the sleeping pills in the bottle.

When Montag stumbles upon his wife's empty bottle of sleeping pills, he goes into a panic and immediately calls for help. Still feeling love for his wife, Montag is concerned for her well-being and is glad to see her rescued in time by the seemingly indifferent men who operate the machine that saves her. Although employed as a fireman, Montag is an empathetic soul that cares deeply about the well-being of others; as a consequence, he is initially panicked and horrified that his wife Mildred might die.
Mildred's suicide attempt, however, serves as a sort of breaking point for Montag. He begins to see his marriage to Mildred as the illusion that it is, and he longs for his companionship with Clarisse. Questioning Mildred's sanity and happiness, he begins to ask her about why she might try to kill herself. Mildred cannot seem to remember the event at all, bringing Montag to question her mind and, as a result, society as a whole.

How is constitutional monarchy different from an absolute monarchy?

There are differences between an absolute monarchy and a constitutional monarchy. In an absolute monarchy, the monarch has total power. The monarch can make laws, determine if a war will occur, and handle relations with other countries. The monarch is not elected and power usually remains within the monarch’s family for many generations.
In a constitutional monarchy, the power of the monarchy is limited. In some cases, the role of the monarch may be ceremonial. In a constitutional monarchy, the citizens of the country elect leaders to make laws. The head of the lawmaking branch is also elected and holds real political power. For example, Great Britain has a constitutional monarchy. The Parliament makes the laws. The members of House of Commons are elected, and the Prime Minister comes from the majority party in the House of Commons.  The Queen appoints the members of the House of Lords. The Prime Minister holds real political power. The role of Queen Elizabeth is mainly a ceremonial role. In a constitutional monarchy, the monarch’s power is limited by a constitution, unlike in an absolute monarchy where no constitution exists.
There are differences between an absolute monarchy and a constitutional monarchy.

Why did Marissa help Melba when she was attacked by the man in the field?

On her way home one day, Melba is savagely attacked by an irate white man who tries to rape her. He's angry at the landmark decision of the US Supreme Court in Brown v. Board of Education (1954), which ruled that school segregation was unconstitutional. Fortunately, Marissa is one hand to help. She hits the man over the head with her book bag and tells Melba to run.
Marissa's actions are somewhat surprising. At school, she has a deserved reputation as a bully. In fact, she bullies Melba on a regular basis, forcing her to give up food and money—despite her father being a well-to-do minister. But here she shows no hesitation in saving Melba from a terrifying ordeal. No explicit reasons are given, but it's more than likely that Marissa intervened because she didn't want to see someone from her own community become the victim of an act of racial and sexual aggression.

College Algebra, Chapter 3, 3.7, Section 3.7, Problem 24

If $g(x) = x^2 + 4x$ with $x \geq -2$, find $g^{-1} (5)$.

To find the inverse of $g(x)$, write $y = g(x)$


$
\begin{equation}
\begin{aligned}

y =& x^2 + 4x
&& \text{Model}
\\
\\
y + 4 =& x^2 + 4x + 4
&& \text{If we solve for $x$, we use completing the square: add } \left(\frac{4}{2} \right)^2 = 4
\\
\\
y + 4 =& (x + 2)^2
&& \text{Perfect Square}
\\
\\
\pm \sqrt{y + 4} =& x + 2
&& \text{Take the square root}
\\
\\
x =& -2 \pm \sqrt{y + 4}
&& \text{Subtract } 2
\\
\\
y =& -2 \pm \sqrt{x + 4}
&& \text{Interchange $x$ and $y$}
\\
\\
\text{Thus,} &
&&
\\
\\
g^{-1} (x) =& -2 \pm \sqrt{x + 4}
&&
\\
\\
\text{Thus, } g^{-1} (5) =& -2 + \sqrt{5 + 4} \qquad \text{ and }
&& g^{-1} (5) = -2 - \sqrt{5 + 4}
\\
\\
=& -2 + 3
&&= -2 - 3
\\
\\
=& 1
&&= -5



\end{aligned}
\end{equation}
$


Last, we have restrictions $x \geq -2$, so..

$g^{-1} (5) = 1$

What are some of the foreshadowed events in "A Good Man Is Hard to Find" by Flanner O'Connor?

The main foreshadowed event in this story is the family's--and most particularly--the Grandmother's encounter with the Misfit, who is mentioned by the Grandmother in the first paragraph of the story. The family's death is another event foreshadowed by the mention of the Misfit. The Grandmother will mention him a second time--quite a lot for a story of this length--when the family stops for lunch at the diner while traveling toward their vacation destination. By the time the car ends up in a ditch in a deserted area and the Misfit and his men appear, we as an audience are not surprised to see him, unsettling as this may be, and not surprised that his intentions are malevolent. Further, since the Grandmother has been the one most obsessed with him, we are not surprised that they have the most interaction, and at the very end, a brief moment of connection. 

How did Renaissance artwork look different from artwork during the Middle Ages?

Because the field of artwork is broad, I will focus on painting. There are many ways in which medieval painting differed from that of the Renaissance, which is what makes the latter period so important as an era of innovation.
First, nearly all medieval painting is religious in its subject matter. A great deal of Renaissance painting is, too, but more liberties are taken in the portrayal of the icons. Renaissance painters individualized the figures, allowed them to emote, showed the natural movement of the body, and often depicted subjects according to classical standards of beauty. Medieval iconography is less inspired and quite redundant. 
Moreover—and this leads to the second major difference—there is no depiction of perspective in medieval painting. The figures appear flat and two-dimensional. Vanishing point perspective developed during the Renaissance. With this technique, a set of parallel lines are perpendicular to a picture plane, drawing one's eye to a single point in space, then outward. This allowed for the illusion of depth on a flat surface.
The depiction of children is another very important difference. Both medieval and Renaissance painters were interested in the depiction of Madonna and Son. Based on the images, however, one thinks that medieval painters had no concept of what children actually looked like. They are little adults: elongated and earnest in their expressions. Renaissance artists painted babies as they are in real life: chubby, playful, and fidgety.

What does Motivation mean?

Asking "what is Motivation?" is equal to asking about happiness, satisfaction, and even sadness. It is entirely up to the individual's life experiences. Evidence of this is the fact that there are a myriad of theories, not "laws," that define motivation. As such, it remains a philosophical debate.
The Dual Factor theory by Hertzberg
This theory says that there are two ways to be motivated: One, is the basic factor of liking what you are doing, or your environment, your co-workers, and so on. Those are, what he calls, the motivating.
The second, is what he calls the hygiene factors. This entails the opposite of the motivating factors. These are the factors that, if they are not good enough, they will be "cleaned out" by the individual. Examples are: salary, benefits, policies, and people.
Maslow's Hierarchy of Needs
This theory dates back to a 1943 paper, written by Abraham Maslow. While motivation is nowhere in the hierarchy, Maslow states that, in order to do anything, one must have, a) safety, b) security, c) health. Those would be the motivating factors to exert a new action. In a way, motivators equate "needs." We need motivation first, in order to conduct an action. Motivation is everything.
Expectancy
This motivation theory relies on basic conditioning. We behave however it fits us best. The motivation is the outcome: for example, if you want to make money, then work harder. If you want to earn the trust of your boss, stay later at work and develop good listening skills in the event that he or she needs to make a friend.
The three factors of motivation in this theory are: expectancy, instrumentality, and valence. In the example given above, the expectancy would be the trust of your boss. The instrumentality would be you, staying later at work to seize the chance to be a sounding board to your boss if he or she needs support. The valence is the end-result: to earn their trust.
As there are other theories (check links included), you will find that the main factors are that a) we need to be of sound of mind and body, b) we need to have a goal in mind, and c) we need to have a way to get it. That is, basically, what motivation means.
http://www.analytictech.com/mb021/motivation.htm

https://lingtechguistics.com/2015/02/05/motivation-and-learning-theories-for-sla-through-chomsky-bandura-vygotsky-krashen-and-gardner/

Single Variable Calculus, Chapter 6, 6.5, Section 6.5, Problem 4

Determine the average value of the function $f(x) = x^2 \sqrt{1 + x^3}$ on the interval $[0,2]$.


$
\begin{equation}
\begin{aligned}

f_{ave} =& \frac{1}{b - a} \int^b_a f(x) dx
\\
\\
f_{ave} =& \frac{1}{2 - 0} \int^2_0 x^2 \sqrt{1 + x^3} dx
\\
\\
\text{Let } u =& 1 + x^3
\\
\\
du =& 3x^2 dx

\end{aligned}
\end{equation}
$


Make sure that your upper and lower limits are also in terms of $u$.


$
\begin{equation}
\begin{aligned}

f_{ave} =& \frac{1}{2} \left( \frac{1}{3} \right) \int^{1 + (2)^3}_{1 + (0)^3} u^{\frac{1}{2}} du
\\
\\
f_{ave} =& \frac{1}{6} \int^9_1 u^{\frac{1}{2}} du
\\
\\
f_{ave} =& \frac{1}{6} \left[ \frac{u^{\frac{3}{2}}}{\displaystyle \frac{3}{2}} \right]^9_1
\\
\\
f_{ave} =& \frac{2}{18} \left[ 9^{\frac{3}{2}} - 1^{\frac{3}{2}} \right]
\\
\\
f_{ave} =& \frac{26}{9}

\end{aligned}
\end{equation}
$

Based on "The Kite Streamers" by Wang Meng, discuss one theme that is related to the Cultural Revolution and one theme about a more universal topic such as love, family relationships, or identity. Also discuss how these two themes are developed throughout the story.

"The Kite Streamers" tells the love story between a young couple named Susu and Jiayuan. It is set in Beijing in the late 1970s at the end of the Cultural Revolution. Although their relationship faces challenges from their society, family, and economic conditions, they strive to support each other in the pursuit of their dreams.
The Cultural Revolution advances the major themes in this story. The upheaval has not only crushed the dreams of the younger generation, it has also indoctrinated certain social views among the general public, such as knowledge is useless, love should serve the communist ideal, and humanity is evil in nature. Most things the couple do in this story, however, are against these views. They learn foreign languages, fall in love without investigating each other’s political background, and believe in helping others even if doing so may harm their own interests. This makes life a lot harder for them, but they have conviction that they are doing the right thing.
Another important theme in this story is the power of dreams, which is what the title, “The Kite Streamers,” symbolizes. Susu and Jiayuan share the similar experience of being sent to the rural area to serve the communist cause and returning after many years through great endeavors. Not given the chance to receive a good education, they have been forced to take very basic positions, as Susu is a waitress and Jiayuan is an umbrella mender. Initially, Susu is disappointed and foresees dim prospects, but Jiayuan’s perseverance awakens her desire to chase her dream. This is how the story turns romantic. They come together and encourage each other to study hard and work hard for their dreams, whatever the results may be.

Wang Meng is a prolific author who faced exile and censorship during the Chinese Cultural Revolution. His story "The Kite Streamers" explores a variety of themes within this context, but personal identity and love are perhaps the strongest.
The Cultural Revolution and Identity
"The Kite Streamers" is a tale about the turmoil of the Chinese Cultural Revolution. The main characters struggle to come to terms with the various ways in which their beloved country has changed after the political revolution. While Mao promised progress at the beginning of the revolution, they soon feel the effects of violence, censorship and authoritarianism. Through the lens of the main characters, Wang explores the nature of the Chinese Cultural Revolution as it relates to personal identity. The author himself was exiled as a result of the revolution and forbidden to write during most of this time, which gives the story an intensely personal feel. Set against the backdrop of a nation in which the government places tight restrictions on all expressions of personal identity, including creative endeavors, "The Kite Streamers" offers a powerful examination of the self. Despite the tumultuous nature of the times, Wang explores the idea that it is possible to find your identity even in the face of great censorship and oppression.
The Theme of Love and Relationships
"The Kite Streamers" focuses primarily on a young couple caught in the middle of the Cultural Revolution. They face not only political upheaval in the outside world but significant pressure in their private relationship as well. The young couple begins the story with a sense of idealism and, as they grow both as individuals and as lovers, that naive optimism is tempered by the reality of their situation. The couple is constantly torn between their dreams for the future and the societal expectations that threaten to consume them. This story is told in the stream of consciousness style through an omniscient observer, which allows the reader to gain a unique understanding of how the characters see each other. The theme of love in the Cultural Revolution is also explored through the concept of restriction. The oppression they experience is so severe that they feel there is nowhere safe for them to go together.
https://books.google.com/books?id=cOxLVO1xloQC&pg=PA379&lpg=PA379&dq=wang+meng+kite+streamers&source=bl&ots=ytbh8-gOal&sig=F0jlFJuo71wtf0oNA3wgAK3lWCk&hl=en&sa=X&ved=0ahUKEwjQjM-i2MzQAhVC0WMKHYkkDbs4ChDoAQgaMAA

https://www.nytimes.com/1986/06/26/world/writer-exiled-by-mao-now-gets-top-culture-post-he-didn-t-want.html

I have been asked to write a short story focusing on one of the characters in Macbeth. I would like to choose a woman as I feel that women often took the backseat in many of Shakespeare's plays. I wanted to write about someone other than Lady Macbeth as she seems like the obvious choice. Does anyone know of any other influential women in the play than I could write about?

Lady Macduff is an interesting character, although she only appears in the one scene when the soldiers invade her castle at Fife. That is in Act IV, Scene 2. You might be able to make an exciting short story out of Lady Macduff. She goes through a lot of stress in a very short time but manages to act with courage and dignity. She tries to carry on a conversation with her little boy in spite of her terrible anxiety and distractions. She doesn't want her son to be affected by her fears. First she is anxious because her husband has fled to England. Then she is warned of impending danger by Ross. Finally a soldier appears with orders to murder her and her son. One of her most striking observations is the following:

Whither should I fly?I have done no harm. But I remember nowI am in this earthly world, where to do harmIs often laudable, to do good sometimeAccounted dangerous folly.

The three witches are also women. It would be quite a challenge to write a short story about one or all of them. 
And then there is the Waiting Gentlewoman who is watching over Lady Macbeth during the famous sleepwalking scene in Act V, Scene 1. The advantage of writing about this scene would be that you could include two women in your story: the Waiting Gentlewoman and Lady Macbeth.

College Algebra, Chapter 1, 1.5, Section 1.5, Problem 64

Find all solutions, real and complex of the equation $\displaystyle x^4 - 16 = 0$


$
\begin{equation}
\begin{aligned}

x^4 - 16 =& 0
&& \text{Given}
\\
\\
(x^4 + x^3) + (x^2 + x) =& 0
&& \text{Group terms}
\\
\\
x^2(x^2 + x) + (x^2 + x) =& 0
&& \text{Factor out } x^2
\\
\\
(x^2 + 1)(x^2 + x) =& 0
&& \text{Factor out } x^2 + 1
\\
\\
x^2 + 1 =& 0 \text{ and } x^2 + x = 0
&& \text{Zero Product Property}
\\
\\
x =& \pm \sqrt{-1} \text{ and } x (x + 1) = 0
&& \text{Solve for } x
\\
\\
x =& \pm \sqrt{i^2} \text{ and } x = 0, x = -1
&& \text{Recall that } i^2 = -1
\\
\\
x =& \pm i \text{ and } x = 0, x = -1
&&


\end{aligned}
\end{equation}
$

A geometric series has third term 36 and sixth term 972. a) find the first and the common ratio of the series.I am able to solve this one which I got a = 4 and r = 3. My problem is in b): the n-th term of the series is U_n, (i) show that sum_(n = 1)^20 U_n = K(3^(20) - 1) where K is an integer to be found. How do I show this?

Hello!
I agree with your answer to the part a), the only possible series is  U_n = 4*3^(n-1).
The question b) becomes simple if we recall the formula of the sum of N terms of a geometric progression U_n with the common ratio r:
sum_(n = m)^(m+N) U_n = ((U_((m+N))) - (U_m))/(r-1).
I give the more general form of the common formula because sometimes there is a confusion related with the starting index of the sum (0 or 1). In this from, the sum is
((the last summed up term of the series) - (the first summed up term of the series)) above (the common ratio - 1).
 
We know already that  U_n = 4*3^(n-1),  m = 1 and N = 20.
Therefore  U_(m+N) = U_21 = 4*3^(21 - 1) = 4*3^20  and the sum is equal to
(4*3^20 - 4*3^0)/(3 - 1) = 4/2 * (3^20 - 1) = 2*(3^20 - 1).
 
Hence the statement we need to prove is true and  K = 2. This is the answer.
https://www.purplemath.com/modules/series5.htm

What is Chapter 10 of Cheaper By the Dozen about?

Chapter 10 of Cheaper By the Dozen is, for the most part, about health and doctors.  Like most of the chapters in the book, it centers on the eccentricities of Mr. Gilbreth and on his obsession with time management studies.
The main “action” in this chapter has to do with the Gilbreth children getting the measles and also getting their tonsils removed.  One major theme of the chapter is Mr. Gilbreth’s attitude towards his children’s health.  His approach is to simply demand that they not be sick.  At the beginning of the chapter, he is quoted as telling his children

You've been given health, and it's your job to keep it. I don't want any excuses. I want you to stay well.

This is very much in keeping with Mr. Gilbreth’s tendency to be (at least seemingly) domineering and demanding when he deals with his children.
However, the chapter also shows us that Mr. Gilbreth is really not a bad or domineering man, no matter what he tries to pretend.  When the children (almost) all get measles, he paints red spots on his face to try to make it look like he has measles too because he misses being around them.  When he gets his tonsils out, he falls apart (after telling the kids there is nothing to the procedure) and has to be nursed back to health for weeks.  These things show how Mr. Gilbreth likes to bluster and pretend to be very severe even though he really is not like that.  This is one of his eccentricities.
The other major theme of the chapter is Mr. Gilbreth’s fascination with trying to make everything as efficient as possible.  We are told that he believes that he could make surgeons more efficient and has filmed an operation.  He critiques the movements of the physicians, talking about how much more efficiently they could be working.  He even tries to film his children’s tonsillectomies, but is foiled when his cameraman forgets to take the lens cap off. 
Thus, Chapter 10 of Cheaper By the Dozen is largely about health, but it is also meant to emphasize the main topics of the book, which are Mr. Gilbreth’s eccentric ways and his fascination with efficiency.

In Seedfolks by Paul Fleischman, what does Ana assume about Kim when she sees her burying her seeds? How does Ana feel about digging up Kim’s beans?

In Chapter 2, Ana sees a girl with dark hair digging into the ground behind an old refrigerator in the lot on Gibb Street. Ana used to be a secretary for the Police Department and immediately becomes suspicious. She thinks that the girl is burying drugs, money, or a gun. Ana mentions that she thought about calling the police but decided to investigate herself. When Ana walks behind the refrigerator and begins to dig in the same location, she ends up digging up the bean seeds that Kim planted. Ana is shocked and upset at herself for digging up Kim's beans. Ana mentions that she felt like she read the girl's secret diary and had ripped out a page without meaning to. She then replants the beans and fills the soil back over them.

Do you capitalize certain types of nouns?

I think we capitalize the proper nouns because we want them to stand out more on the page, and we also want to accord proper nouns a certain amount of respect and attention.  When we look at the word "house," for example, there is no good reason to call special attention to it, but if we have a title like "Chicago Settlement House," we want to give it the extra attention accorded to a particular place.  Capital letters make a word stand out much more easily, certainly.  In looking at directions, when we capitalize the name of a street, it stands out more for the reader.  Another example is the word we often use for a mother.  That is a generic term, for any mother, but in dialogue, we write like this:

I see, Mom, how proud you are of me.

Otherwise I would write this:

My mom is proud of me.

So there is some importance, some respect, and some special attention involved. 
 

"Noun" is the designation for a grammatical concept, a type of word. A noun is generally described as a "person, place, or thing," but it also includes ideas, such as democracy. Verbs designate action words but nouns are the entities that do the actions or have the actions done to them. 
There are two types of nouns: common nouns and proper nouns. Common nouns are the kinds of nouns that are in the sentences of this answer, e.g., words such as "noun," "concept," "type," and "word." These are generic. There are many nouns, many concepts, many types of things, and very many words!
Proper nouns, in contrast, designate specific things or ideas. You can have a city (common noun) or you can have "New York," which is a specific city and a proper noun. There can be a person, or you can have "Barak Obama," as a specific person and also a proper noun. 
Proper nouns, as can be seen from the examples above, are the type of nouns that are capitalized always. 
http://www.chompchomp.com/terms/propernoun.htm

How are women's roles different from men's roles in the novel? What does Esperanza believe the role of women in society should be instead?

In The House on Mango Street, women's roles seem to be submissive under a patriarchal society. Many of the vignettes focus on women who were married young, have children, and are either beaten or abandoned by their husbands. Luckily for Esperanza, her father treats her mother with great respect, and she has a good home; however, she notices those women just a few years older than she is and isn't impressed. For example, in the third vignette, entitled "Boys and Girls," Esperanza explains the following about her culture: "The boys and the girls live in separate worlds. The boys in their universe and we in ours" (8). Clearly, Esperanza discovers this division between genders early on and then starts looking outside of the world of children to discover just how these two different universes exist in adults. One example of how women view men in Esperanza's world is through Marin's eyes, as follows:

"Marin says that if she stays here next year, she's going to get a real job downtown because that's where the best jobs are, . . . and [she] can meet someone in the subway who might marry you and take you to live in a big house far away" (26).

Basically, Marin is telling Esperanza that marriage is the way to get out of poverty and that she'll be able to do it because she is beautiful. Unfortunately, Esperanza doesn't feel beautiful in her own eyes. Then, as the book progresses, she discovers that men can not only be creepy and gross, but also take advantage of and abuse the women in their lives. As a result, she loses trust in men and decides to plan a life for herself without a man. Esperanza explains as follows:

"I am an ugly daughter. I am the one nobody comes for. . . . I have begun my own quiet war. Simple. Sure. I am one who leaves the table like a man, without putting back the chair or picking up the plate" (88–89).

She will wage a "quiet war" because she doesn't want to upset anyone or call any attention to herself, but she plans to never allow what has happened to other women in the novel to happen to her. Furthermore, Esperanza points out that men are disrespectful to women because they leave their chairs out and don't clear their plates after eating a dinner that was most likely made by a woman. They do these things because they believe that women should serve men, stay home, and have babies. Consequently, Esperanza decides that one day she will own a house of her own and obtain it through her own efforts, not depending on a man to get it for her.

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 50

Prove $\displaystyle \int \sec^n x dx = \frac{\tan x \sec^{n-2} x}{n - 1} + \frac{n-2}{n-1} \int \sec^{n-2} x dx ( n \neq 1)$
Notice that $\displaystyle \int \sec^n x dx = \int \sec^{n - 2} x \cdot \sec^2 x dx$

So if we let $n = \sec^{n-2} x (\sec x \tan x) dx$ and $dv = \sec^2 x dx$, then

$du = (n-2) \sec^{n-2} x ( \sec x \tan x ) dx$ and $v = \int \sec^2 x dx = \tan x$

Thus,

$
\begin{equation}
\begin{aligned}
\int \sec^{n-2} x \cdot \sec^2 x dx = uv - \int v du &= \tan x \left( \sec^{n-2} x \right) - \int (\tan x) \left[ (n-2) \sec^{n-3} x (\sec x \tan x) \right]\\
\\
&= \tan \left( \sec^{n-2} x \right) - (n-2) \int \sec^{n-2} x \tan^2 x
\end{aligned}
\end{equation}
$


Recall for the identity $\tan^x = \sec^ x - 1$
So,
$\displaystyle [(n-2)+1] \int \sec^{n-2} x \cdot \sec^2 x dx = \tan x \left( \sec^{n-2} x \right) + (n-2) \int \sec^{n-2} x dx$
$\displaystyle (n-1) \int \sec^{n-2} x \cdot \sec^2 x dx = \tan x \left( \sec^{n-2} x\right) + (n-2) \int \sec^{n-2} x dx$

Dividing with sides by $(n-1)$, we get
$\displaystyle \int \sec^{n-2} x \cdot \sec^2 x dx = \frac{\tan x \left(\sec^{n-2}x\right)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x dx$

Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 86

Evaluate the equation $|3x - 1| = |3x + 9|$.

This equation is satisfied either if $3x - 1$ and $3x + 9$ are equal to each other or if $3x - 1$ and $3x + 9$ are negatives of each other


$
\begin{equation}
\begin{aligned}

3x - 1 =& 3x + 9 && \text{or} &&& 3x - 1 =& -(3x + 9)
\\
3x - 3x =& 9 + 1 && \text{or} &&& 3x + 3x =& -9 + 1
\\
0 =& 10 && \text{or} &&& 6x =& -8
\\
& && \text{or} &&& x =& - \frac{4}{3}

\end{aligned}
\end{equation}
$


The solution set is $\displaystyle \left \{ - \frac{4}{3} \right \}$.

Who does Ponyboy blame for everything that goes wrong in chapter 4? What is Dally's reaction to this and how does it change Ponyboy's perspective?

Chapter 4 is the chapter that has Ponyboy and Johnny getting attacked by the Socs. The rival gang is drowning Ponyboy, and Johnny kills Bob in order to save Ponyboy. Johnny tries to take a lot of the blame in a later chapter, but Ponyboy doesn't accept that at all.

"This is my fault," Johnny said in a miserable voice.

While Johnny might have killed someone, Ponyboy doesn't blame Johnny for anything. Johnny acted in defense. Nothing would have happened had the Soc gang members not attacked Ponyboy and Johnny. The Socs came looking for a fight too because the park was in Greaser territory.
Dally's overall reaction is a reaction of calm and calculating. Dally is no stranger to breaking the law, and he's actually proud of Johnny for what he did.

He didn't bat an eye when Johnny told him what had happened, only grinned and said "Good for you" when Johnny told how he had knifed the Soc.

Because of Dally's experience with breaking the law, he immediately understands the severity of the situation that Ponyboy and Johnny are in. Dally knows that "murder" is a word that is now going to be linked with the boys; therefore, the boys have to hide out for awhile. Dally's use of that word and seriousness indicate to Johnny and Ponyboy that they could be in deeper trouble than they initially thought.

In chapter 4, Johnny ends up stabbing Bob the Soc to death after his gang attacks the Greasers and tries to drown Pony. The Socs attack Johnny and Pony as they rest at the park. The park is in the Greaser’s territory. Everything happens so fast that Pony realizes what has happened after he sees Bob’s body on the ground.
The boys are forced to run. Pony blames the entire incident on the Socs because they were trespassing and looking for trouble. Johnny had no choice but to react in self-defense and save Pony from being drowned. However, after they meet with Dally, he reminds them that the situation they are dealing with involved murder. Although Dally reacts calmly, both Pony and Johnny understand that they are up against a very serious issue. At that point, Pony is worried that he might not see his brothers or friends ever again because those involved in the incident will have to run and hide for the rest of their lives.

What is Aunt Alexandra's vision for what is "lady like"?

Aunt Alexandra represents the genteel South. While her brothers are easy going and laid back, as a woman, Aunt Alexandra doesn't have that luxury. She is stern and holds the values of the South in the highest regard. She tries to impart these in Atticus's children, though they challenge her more traditional ways.
When it comes to Scout, Aunt Alexandra's expects her to behave like a lady, not a child. When Scout wants to play with Walter, Aunt Alexandra prohibits it. She responds to Scout's request by saying, "I'll tell you why," she said. "Because—he—is—trash, that's why you can't play with him. I'll not have you around him, picking up his habits and learning Lord-knows-what."(23.86-87)
In addition to the value Aunt Alexandra places on lady-like, genteel behavior and her tacit disapproval of the way Atticus is raising his children, Aunty values the family name above all else. Scout's lady-like behavior, or her lack of it, affects the family name. We see this when she discusses other local families:
Aunt Alexandra, in underlining the moral of young Sam Merriweather's suicide, said it was caused by a morbid streak in the family. Let a sixteen-year-old girl giggle in the choir and Aunty would say, "It just goes to show you, all the Penfield women are flighty." Everybody in Maycomb, it seemed, had a Streak: a Drinking Streak, a Gambling Streak, a Mean Streak, a Funny Streak. (13.26)
Auntie's vision for what is lady-like, then, relies on the traditions of the past and on upholding the family honor.

What the cat symbolizes in "The Blue Bowl"?

"The Blue Bowl" by Jane Kenyon does not seem to be a highly symbolic poem, but the way the family handles the burial of the cat could be said to represent how people deal with all sorts of losses. 
The first eight lines of the poem read,

Like primitives we buried the cat
with his bowl. Bare-handed
we scraped sand and gravel
back into the hole. It fell with a hiss
and thud on his side,
on his long red fur, the white feathers
that grew between his toes, and his
long, not to say aquiline, nose (1-8).

Kenyon's speaker describes how she and her family members buried their dead cat with its bowl as being "like primitives." This seems to reference ancient cultures that believed they should bury the dead with objects that they could use in the afterlife. It is also "primitive" that they use their hands to cover the hole. The cat is then described in some detail, but it is the bowl that falls "with a hiss," as the cat can no longer make such noises. Kenyon's choice to start the poem with "like primitives" sets up the idea that the family's feelings about the loss of the cat are universal and instinctual. 
 
To describe the family's feelings and behaviors after burying the cat, the speaker continues, 


We stood and brushed each other off.
There are sorrows much keener than these.
Silent the rest of the day, we worked,
ate, stared, and slept. It stormed
all night; now it clears, and a robin
burbles from a dripping bush
like the neighbor who means well
but always says the wrong thing (9-16).



When the speaker says, "We stood and brushed each other off," she seems to suggest that the burial has allowed the family to bond and also that, symbolically, humans help each other to recover in times of loss. The speaker next acknowledges that there are losses greater than that of a family cat; however, the family's behavior during the rest of the day and the closing thought of the poem suggest that losing a pet can have as significant an impact as any other death. The family goes through their routine, but there is no energy in their movements. Then, after it storms all night, a bird sings in the morning, as though the pain is over and everything is better now. Of course, that is not the case; the bird is compared to "the neighbor who means well / but always says the wrong thing." The bird cannot understand that the family still feels the loss of the cat, just as those outside the community that is grieving cannot truly know how those inside feel. The singing is a nice gesture, but it does not make the family forget about the death of the cat. Yes, life goes on, but everyone moves on at a different pace. 
 
Ultimately, the loss of the cat allows Kenyon's speaker to reflect on universal reactions to loss and the feelings and behaviors that are part of its aftermath. 

In The Old Man and the Sea, different readers may place the climax at different points. Where do you think the climax is in this novel? Defend your position using evidence from the novel.

The climactic action of The Old Man and the Sea involves Santiago's ability to successfully pull in the great marlin and tie him to his boat, as well as his brave battle against the sharks who would deprive him of his success.
Because an enduring work such as Hemingway's tale lends itself to extended interpretations and broader meanings, the climax of the narrative is sometimes interpreted as other than the point at which Santiago finally conquers the marlin by impaling it with his harpoon. Some interpretations of this work extend the climax to include Santiago's heroic struggle against the sharks who would deprive him of the great fish. 
With the definition of the hero as,

A man who lives correctly, following the ideals of honor, courage, and endurance in a world that is sometimes chaotic, often stressful, and always painful, 

it seems that the climax should include Santiago's enduring struggle against the marlin and the sharks. He never gives up in his effort to bring back the marlin. Although he loses the marlin's meat to the sharks, Santiago finds the inner strength to continue to endure his greatest struggle. There is an extended climax because the high emotion remains as Santiago fights against the sharks for the same reason that he fights for the marlin. This reason for Santiago's fighting is what one critic calls "the intangibles that can redeem his individual life."
 At one point in his struggles, the old man tells himself,

You did not kill the fish only to keep alive and to sell for food, he thought. You killed him for pride and because you are a fisherman.

It is because of his pride as a fisherman that Santiago continues his battle by fighting the sharks. He repeats to himself, "Fight them. . . I'll fight them until I die." This statement also indicates that the narrative is still at a point of high emotion, a climactic point. 
The falling action occurs when Santiago returns to the little harbor, and he carries his mast over his shoulder up the hill. He is so exhausted by his fight that he must rest five times. Nevertheless, Santiago arrives home, lies on his bed, and falls asleep. As he sleeps, the old fisherman can dream since he has met all the challenges presented to him. Because of his endurance against both the marlin and the sharks, along with his perseverance during these climactic moments, Santiago can retain his pride and know that he is still a man. 

lim_(x->0) sin(5x)/tan(9x) Evaluate the limit, using L’Hôpital’s Rule if necessary.

lim_(x->0) (sin (5x))/(tan(9x))
To solve, plug-in x = 0.
lim_(x->0) (sin(5*0))/(tan(9*0)) = 0/0
Since the result is indeterminate, to find the limit of the function as x approaches zero, apply L'Hopital's Rule. To do so, take the derivative of the numerator and denominator.
lim_(x->0) (sin(5x))/(tan(9x)) = lim_(x->0) ((sin(5x))')/((tan(9x))') = lim_(x->0) (5cos(5x))/(9sec^2(5x))
And, plug-in x =0.
= (5cos(5*0))/(9sec^2(5*0) )= (5cos(0))/(9sec^2(0)) =(5*1)/(9*1)=5/9
 
Therefore,  lim_(x->0) (sin (5x))/(tan(9x))=5/9 .

Why does Jaggers call Wemmick the "most cunning imposter in all of London"?

In chapter 51 of Great Expectations Pip wants Jaggers to be more forthcoming about the identity of his benefactor, not just for his own sake, but Estella's. Pip is sure that he already knows the truth, but wants to hear it from the horse's mouth, as it were. But initially it seems that Jaggers is reluctant to divulge the full story, so Pip turns to Wemmick, who's also present, to intercede on his behalf.
In making his appeal, Pip refers to Wemmick's kind heart, and also to his father, the implication being that he's a good son who dotes on this senile old man. Upon hearing this, Jaggers is rather taken aback. This isn't the Wemmick he's come to know; the Wemmick who works in his office; the Wemmick who can so often be such a calculating individual in his business dealings. That's when Jaggers jokingly refers to Wemmick as the most cunning impostor in all London. It's as if he's been leading a double life all these years.

In On the Road, in what ways does alcohol represent Sal's need for new experiences?

Alcohol plays a big role in the work of Jack Kerouac, and On the Road is no exception. One of the most prominent themes in this novel is the importance of having new experiences, altering one's perception of the world, and generally having a good time. Alcohol plays a central role in these enterprises for a couple of reasons. First, and perhaps most simplistically, alcohol use helps characters in the story supposedly "have a good time" (at least, until their alcohol-fueled adventures result in the seriously depressing dissolution of their relationships). More importantly, however, alcohol serves as a tool for altering one's perception of reality, as alcohol consumption changes one's state of consciousness and leads one to see the world in a different way. Thus, drinking alcohol ostensibly facilitates the characters' ability to have new experiences, as it enables them to engage with the world with a different consciousness. Additionally, Sal's reliance on alcohol represents his need for new experiences because it proves his need for an altered perception (i.e., a new way of experiencing the world). However, it's important to note that, later on in Kerouac's life, his use of alcohol descended into alcoholism, and much of his later work (such as his novel Big Sur) focuses less on the "benefits" of altered perception, and more on his struggles with alcohol dependency and his desire to become sober.  

What is an example of women being objectified and/or gender stereotypes in Macbeth?

An often overlooked character in Macbeth is Macduff's wife. She can be viewed as a foil to Lady Macbeth. While Lady Macbeth shirks the role of meek and dutiful wife and seeks power through demonic means, Macduff's wife embodies the stereotype of a docile, passive woman. For this, she pays the ultimate price, but through her character we can see the limited agency available to women in the play and in the culture it is a part of.
Notably, Macduff's wife does not have a name. Similar to Lady Macbeth, she is known by her husband's name only, but in the case of Macduff's wife she does not even have a title. In Act 4 Scene 2, after Macbeth learned that Macduff has fled to England and has ordered the murder of his family, Macduff's wife laments her abandonment: "To leave his wife, to leave his babes... He loves us not." She is afraid that, whether guilty or not, Macduff's fleeing to England makes him appear guilty, and puts her family at risk for retaliation. To her, he left them behind, unguarded and vulnerable. As a woman, she has very little power to protect herself.
Even her son is aware that she is vulnerable without a man's protection, asking her, "Nay, how will you do for a husband?" Even as young as he is, the boy recognizes that his mother alone is in a dangerous position.
Shortly after, a messenger arrives to warn her that a murderer is approaching her household. Macduff's wife has no time to act and no protection--first her husband has left, then Ross, and then the messenger. She is alone, with no where to go. She knows she is innocent, but that bad things still happen to innocent people, while evil people are often rewarded. At the very end, she questions the virtue of being a meek and virtuous woman: "I am in this earthly world, where to do harm is often laudable, to do good sometime accounted dangerous folly. Why, then, alas, do I put up that womanly defense, to say I have done no harm?"
Macbeth might be read through a lens that questions the female stereotype of docility and obedience. Both Lady Macbeth and Macduff's wife meet untimely ends. One is a victim of power, the other abuses power. But both women demonstrate the limited ability of women to access power, and suggest that, in a patriarchal society, it might only be attained for women through demonic means.

One of the best examples in Macbeth of gender stereotypes, specifically the way in which women were expected to behave, comes when Lady Macbeth receives the message that Macbeth is returning home in act 1, scene 5. It is evident from Lady Macbeth's speech here that she—and, by analogy, the Shakespearean audience—believe that the cruel and ambitious parts of her nature are unwomanly and not to be expected in a woman. So strongly does she feel this that she calls upon the "spirits" to "unsex me here," or divorce her from her feminine nature so that she might have the strength to shore up Macbeth's ambition. She feels that her husband will not, alone, have the will to pursue his goals, and therefore she is determined to "pour [her] spirits in [his] ear."
In order to do this, then, Lady Macbeth must become something crueler and more lacking in compunction than any woman should be, according to the gender stereotyping of the time. She calls to the spirits to fill her with "direst cruelty," stemming any "passage to remorse." The direct connection between this and the removal of femininity is made evident in her demand that the "murdering ministers" take the milk "for gall" from her "woman's breasts."
Later, this idea is revisited when Lady Macbeth is encouraging her husband to "screw [his] courage to the sticking-place" by questioning his masculinity. When he dared kill Duncan, she says, "then you were a man." If he hesitates in this pursuit, his weakness makes him less of a man. Lady Macbeth demonstrates her own fixedness of purpose by stating that had she believed Macbeth would be so hesitant she would have "pluck'd [her] nipple from" the "boneless gums" of her own child. Maternal love, one of the key tenets of femininity, is presented in this image: Lady Macbeth is suggesting that for Macbeth to turn away from his purpose is as unmanly as it would be unwomanly for Lady Macbeth to have killed her own child.
Ambition and even cruelty are very clearly identified as masculine traits in this play, with Lady Macbeth becoming "unwomanly" in the characteristics she embodies, while women are believed to be softer, sweeter, and less resolute.

College Algebra, Chapter 4, 4.4, Section 4.4, Problem 82

Determine all rational zeros of the polynomial $P(x) = 8x^5 - 14x^4 - 22x^3 + 57x^2 - 35x + 6$ and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula or other factoring techniques.
The possible rational zeros of $P$ are $\displaystyle \pm \frac{1}{4}, \pm \frac{3}{8}, \pm \frac{1}{2}, \pm \frac{3}{4}, \pm 1, \pm \frac{3}{2}, \pm 2, \pm 3, \pm 6$. We check the positive candidates first, beginning with the smallest
Using Synthetic Division, we have



So, $\displaystyle \frac{3}{4}$ is a zero and $\displaystyle P(x) = \left( x - \frac{3}{4} \right) \left( 8x^4 - 8x^3 - 28x^2 + 36x - 8 \right)$ or $\displaystyle P(x) = 4\left( x - \frac{3}{4} \right)\left( 2x^4 - 2x^3 - 7x^2 + 9x - 2 \right)$. We now factor the quotient $2x^4 - 2x^3 - 7x^2 + 9x - 2$. The possible rational zeros of $P$ are $\displaystyle \pm \frac{1}{2}, \pm 1, \pm 2$. Using Synthetic Division, we get



So, $1$ is a zero and $\displaystyle P(x) = 4\left( x - \frac{3}{4} \right) (x-1) \left( 2x^3 - 7x + 2 \right)$. We now factor the quotient $ 2x^3 - 7x + 2$. We still have same list of possible rational zeros except $\displaystyle \frac{1}{2}$ is elminated. Using Synthetic Division, we get


So, $-2$ is a zero and $\displaystyle P(x) = 4\left( x - \frac{3}{4} \right) (x-1)(x+2) \left( 2x^2 + 4x -1 \right)$. We now factor the quotient $ 2x^2 + 4x -1 $. By using quadratic formula,


$
\begin{equation}
\begin{aligned}
x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\
\\
x &= \frac{-4 \pm \sqrt{(4)^2 - 4(2)(-1)}}{2(2)}\\
\\
x &= \frac{-2\pm\sqrt{6}}{2}
\end{aligned}
\end{equation}
$

Therefore, $\displaystyle P(x) = 4\left( x - \frac{3}{4} \right) (x - 1)(x + 2) \left( x+\frac{2+ \sqrt{6}}{2} \right) \left( x + \frac{2-\sqrt{6}}{2} \right)$
This means that the zeros of $P$ are, $\displaystyle \frac{3}{4}, 1, -2, \frac{-2+\sqrt{6}}{2} \text{ and } \frac{-2-\sqrt{6}}{2}$.

Calculus of a Single Variable, Chapter 7, 7.1, Section 7.1, Problem 5

Given the curve equations ,they are
y_1 = 3(x^3 - x) -----(1)
y_2 = 0 -----(2)
to get the boundaries or the intersecting points of the functions we have to equate the functions .
y_1=y_2
=> 3(x^3 - x)= 0
=> (x^3 - x)=0
=> x(x^2-1)=0
=> x=0 or x=+-1
so,
so,
the Area =int_-1^0 3(x^3)-x) -0 dx + int_0 ^1 0-(3x^3-x) dx
= int_(-1) ^0 (3x^3-3x) -0 dx +int_(0) ^1 0-(3x^3-3x) dx
= [(3x^4)/4 -3/2 x^2]_(-1) ^0 +[-(3x^4)/4 +3/2 x^2]_(0) ^1
=[0]-[3/4 - 3/2] +[-3/4+3/2]-[0]
=-3/4 +3/2-3/4+3/2
= -3/2 +3
=3/2 = 1.5 is the area of the region between the curves

In "Shooting an Elephant" by George Orwell, are the Burmese capable adults who can rule themselves or do they like having the British Empire take care of them, despite their calls for independence?

In "Shooting an Elephant," Orwell's experience of working in Burma suggests that the native people did not like being ruled by the British. We see this through their reaction to Orwell as he carried out his professional duty as a sub-divisional police officer. Buddhist priests jeered at him, for example, and football players tripped him up on the field. This is, arguably, a result of the nature of Britain's rule in Burma. As Orwell comments, British rule was not consensual:

I thought of the British Raj as an unbreakable tyranny, as something clamped down…upon the will of prostrate peoples.

Moreover, this attitude of resentment among the Burmese suggests that they really did want to be independent but were afraid to break Britain's rules. We see this through Orwell's description of the Burmese prisoners in the "stinking cages of the lock-ups" in which prisoners were "bogged with bamboos."
It is, therefore, logical to suggest that the Burmese were more than capable of self-rule but were unable to achieve this because of the strength of British power. It was not until 1948, two decades after Orwell's service, that the British finally returned Burma to its native people. 

What is a theme in Act I, Scene 1 of Arthur Miller's The Crucible?

The fear of "the other" is one of the dominant themes in Act I, Scene 1 of The Crucible.
"The other" can be defined as anything different.  In his extensive stage directions, Miller suggests that the residents of Salem possessed a strong fear of this other. Miller explores this theme in several ways.  One way is in his suggestion that the people of Salem believed their lives should exist only for religious expression:

No one can really know what their lives were like. They had no novelists - and would not have permitted anyone to read a novel if one were handy. Their creed forbade anything re-sembling a theater or “vain enjoyment.” They did not celebrate Christmas, and a holiday from work meant only that they must concentrate even more upon prayer. 

Salemites' clutching to religion meant that anything outside of its jurisdiction was transgression and evil.  The fear of "the other" caused the people of Salem to always pay attention to what other people did. People in Salem clung to conformity as the only weapon against "the other": "But, as in all such matters, the issue was not clear-cut, for danger was still a possibility, and in unity still lay the best promise of safety."  People in Salem took sanctuary in conformity as a defense against anything believed to be different.  
This fear compelled the people of Salem to focus on what other people said and did.  Miller argues that the preoccupation with others was out of fear that they were doing something that the community did not sanction: "This predilection for minding other people’s business was time-honored among the people of Salem, and it undoubtedly created many of the suspicions which were to feed the coming madness." Miller suggests that the fear of the other was what caused Salemites to focus on what other people said and did.  He also believes that it played a significant role in the hysteria behind the trials.  To be accused of being a witch meant that a person was "different" and had broken from the community's expectation of spirituality.  
This theme is established in the play's opening scene.  It is evident in how Parris is worried about what others will say about him regarding the girls' actions in the woods.  It is also seen in how the girls launch into their accusations and how the townspeople respond.  The fear of "the other" theme introduced in the first scene of the first act is a significant theme throughout the drama.

Who begins to pay Kit weekly visits in Chapter Seven of The Witch of Blackbird Pond by Elizabeth George Speare?

In Chapter Seven of Elizabeth George Speare's The Witch of Blackbird Pond, William Ashby visits Kit at her Uncle Matthew and Aunt Rachel's home. Kit finds this encounter utterly boring, as William seems happy to just sit in silence as Kit awkwardly tries to make conversation. William only begins to show some spirit when he gets into a debate with Uncle Matthew over surrendering the charter. 
The night ends on a sour note, with Uncle Matthew storming off in a huff. Kit is certain that William won't be back to visit her, but Judith, Mercy, and Aunt Rachel dismiss that notion by telling her that William has decided to start constructing his home, a sign that he has decided to seek Kit's hand in marriage. This horrifies Kit, who longs for someone that she can talk to.
As Kit's family predicted, William begins to visit Kit every Saturday, and she gradually begins to look forward to these visits, as they break the monotony of her otherwise laborious work week. 

Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 37

The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval.
The mean value theorem states:
f(b) - f(a) = f'(c)(b-a)
Replacing 1 for b and -2 for a, yields:
f(1) - f(-2) = f'(c)(1+ 2)
Evaluating f(1) and f(-2) yields:
f(1) = 1^2 => f(1) =1
f(-2) = (-2)^2 => f(-2) = 4
You need to evaluate f'(c):
f'(c) = (c^2)' => f'(c) = 2c
Replacing the found values in equation f(1) - f(-2) = f'(c)(1 + 2):
1 - 4 = 2c(1+2) => 6c = -3 => c = -3/6 => c = -1/2 in [-2,1]
Hence, in this case, the mean value theorem can be applied and the value of c is c = -1/2 .

log_5sqrt(x - 4) = 3.2 Solve the equation accurate to three decimal places

To simplify the logarithmic equation: log_5(sqrt(x-4))=3.2 , recall the logarithm property:  a^((log_(a)(x))) = x .
When a logarithm function is raised by the same base, the log cancels out which is what we need to do on the left side of the equation.
As a rule we apply same change on both sides of the equation.
 Raising both sides by base of 5:
5^(log_5(sqrt(x-4)))= 5^(3.2)
sqrt(x-4) = 5^(3.2)
To cancel out the radical sign, square both sides:
(sqrt(x-4))^2 = (5^(3.2)) ^2  
  x-4 =5^(6.4)
 x= 5^(6.4)+4
  x~~29748.593  (rounded off to three decimal places)
To check, plug-in x=29748.593 in log_5(sqrt(x-4)) :
log_5(sqrt(29748.593-4))
log_5(sqrt(29744.593))
log_5(172.4662083)=3.2 which is what we want
 
So, x=29748.593 is the real solution.
Note: (x^m)^n= x^((m*n ))
 
 

What are ways "The Deep River" by Bessie Head relates to the theme of identity? Include both personal and cultural identity issues in your discussion.

Head’s story is about identity and individuality. The Monemapee live “with one face” under the rule of their chief. The chief makes all the decisions, such as when to plough, when to harvest, and when to prepare the crops, and the people simply follow his orders. The unity and security of this arrangement is what is meant by the “deep river” of the title—that is, the tribe is at peace and protected as if they were suspended in a deep river. The phrase also suggests another river: the River Lethe, the river of forgetfulness in the Greek underworld. The communal memory of the tribe is reduced to a single word—Talaote. The tribe may act as if they have one face, but “during their journey southwards,” they have forgotten where they came from and what their original language was. 
When the old chief dies, his eldest son, Sebembele, announces that he is in love with the old chief’s youngest wife, Rankwana, and that the old chief’s youngest son, still a baby, is in fact his son, by Rankwana. This causes a kind of crisis: first, Sebembele’s younger brothers see his acknowledgement that the baby is his own as undermining their own positions within the tribe; second, because women are “of no account,” Sebembele’s love for Rankwana is a sign of weakness. Anyone who listens to a woman is “like one who listens to the advice of a child.” Sebembele is advised to pick out another wife, and he is deeply conflicted.
Sebembele cannot bring himself to do this. Instead, he publicly claims Rankwana and her child. They walk together, through the town, Sembembele holding the child in his arms. He became “a ruler who talked with deeds rather than words,” and the people saw that “the time had come for them to offer up their individual faces to the face of this ruler.” His love for Rankwana is a radical act in that he is singling out a woman for affection. This is particularly dangerous in that it recognizes Rankwana as someone worthy of respect and love. It is also an origin story. Sebembele leaves with those that support him, to begin a new tribe, eventually settling “in the land of Bamangwato.” Characteristically, perhaps, the details of the story are not remembered completely by his descendants: the old men give “confused and contradictory accounts of their origins, but they say they lost their place of birth over a woman.” 

In some ways, this story pits personal or individual identity against cultural or communal identity.Long ago, the Talaote tribe "lived without faces," meaning people lived without personal identity, having only tribal or cultural identity. They were content to have their identity be represented by their chief.  Everyone followed the chief's orders, acting as one, accepting "this regimental leveling out of their individual souls" until, one day, conflict came and "the people awoke and showed their individual faces." It is when conflict arrives that we see how individuals can suddenly begin to privilege personal identity over cultural identity; or, perhaps, the emerging importance of personal identity over cultural identity is the cause of the conflict itself?
When Sebembele, the oldest son of the now-dead chief Monemapee, admits that he had an affair with his father's most junior bride, Rankwana, this causes conflict. The affair resulted in Sebembele fathering Rankwana's son, who he refuses to give up. Some tribe members feel he should be able to keep Rankwana as a wife, while others think a man who is so influenced by a woman is unfit to rule. Sebembele is unable to make up his mind, so his brothers choose for him, sending Rankwana away to be married to another man. When Sebembele learns of his brothers' actions, he goes to Rankwana's new home to retrieve her and their son; they walk through the village as one and seem prepared to leave the tribe together when something remarkable happens. Sebembele's supporters are so impressed with his actions that they feel "the time had come for them to offer up their individual faces to the face of this ruler." Sebembele's camp packs up and leaves the tribe, adopting a new name and trading their old cultural identity for a new one.
This story seems to suggest that cultural identity can only successfully take the place of individual identity until the individual begins to chafe under it. If one's identity comes into conflict with one's cultural identity, then personal identity automatically becomes more important, as the people are unwilling to live "without faces" when they disagree with their rulers or the rules. 

What sort of methods/theories did the author employ, and what sort of sources did the author rely on?

Rachel Carson was a biologist who worked for 16 years (1936-52) for the Fish and Wildlife service, so she had a strong—and long—background in the science she documented for her book. She had been worried about the use of pesticides since the 1940s. Because of her concerns, she had made a point of following court cases that fought the use of pesticides. In one case, for example, people in Long Island sued to prevent their land from being sprayed with DDT to control the gypsy moth. 
One theory she relied on was biomagnification, which argued that the cumulative effects of toxins as they build up in the environment and the human body are worse than the sum of their parts. She also relied heavily on the medical field, tying together medical and biological research to argue that pesticides caused cancer.
In the four years before publishing her book, Carson did a vast amount of research. She did much of it at the NIH (National Institutes of Health) and the Library of Medicine. She examined thousands of scientific papers on pesticide use and related issues. She interviewed scientists. She also discovered that medical researchers were studying the link between pesticides and cancers and incorporated their work into her research. She interviewed doctors. She used the work of Wilhelm Hueper, who had determined that many pesticides caused cancer. She also researched history to document her link between toxins and cancer, even using research on cancer in chimney sweeps from the eighteenth century, which connected their scrotal cancer to constant contact with soot. 
Carson's method was not simply to rely on scientific reports and papers that would be difficult for the average person to understand. She used newspaper reports, personal interviews, and court cases to show how pesticides were impacting everyday life. Trying to build sympathy for her case, she showed the devastating effects of pesticides on the beloved robin, and on a living symbol of the United States, the eagle. She also showed how pesticide use had harmful effects on ordinary humans. She likened pesticides to radiation poisoning from atomic bombs: invisible but devastating. That would have been a potent parallel for people still coming to grips with atomic power. She very pointedly aimed her research at persuading people that their own lives were being damaged—or could be damaged very easily—by the overuse of pesticides that entered the food chain and ended up in the human body.

How can I summarize and analyze every stanza of Auden's poem "In Memory of W.B. Yeats"? Who are the "rich women" and the "dogs"?

It sounds like you're writing an essay on this poem, which is a lament, or elegy, by W.H. Auden for the Irish poet W.B. Yeats. It's a big poem, but I'll try to get you started on your answer, and make sure you understand the parts you're unclear on.
The poem is, of course, divided into stanzas, but it is also broken into three sections, which is a more useful way to look at it. Traditionally, all elegies are in three sections: a lament for the dead; a section praising the dead; and a section consoling the reader. Auden's poem follows this structure.
Part I ("he disappeared in the dead of winter") uses pathetic fallacy to demonstrate how all of nature mourned the "dark cold day" of Yeats's death. The cold and the snow that lay on earth represent the feeling of coldness that came over all as a reaction to Yeat's death.
There is an element of consolation creeping into this section—"the wolves ran on through the evergreen forests"—suggesting that not the entire world was affected by this one death. But Auden refocuses the reader on the poem's subject: "for him it was his last afternoon as himself." He uses metaphor to depict Yeats as a city whose "provinces" revolted and "squares became empty." As he died, "silence invaded the suburbs" and the city of the poet was emptied.
Instead of being contained as himself, then, Yeats is now "scattered among a hundred cities," seemingly a suggestion that the constitutent parts of his soul have been divided up among others. He "became his admirers" in that all he believed lives on only in them, now, not in him. Auden uses anaphora to enumerate the many who will "think of this day" in the future as a sad day, the repeated "and" serving to demonstrate just how many lament Yeats's death.
Part II is notably brief. It describes Yeats succinctly as "silly like us," and emphasizes the key point that "your gift survived it all." When Auden mentions "the parish of rich women" here, he is referring to the upper class women (grouped as a "parish") who misinterpreted or misunderstood Yeats, particularly those in "mad Ireland" who could not understand Yeats's position on the Irish question. Yeats's poetry, Auden says in this stanza, was caused by his beliefs, by his "hurt," and will survive him; but it is not poetry that makes things happen, but the things that happen to us which make poetry happen.
In Part III, the consolation, the structure and meter echoes a funeral eulogy or memorial poem on a gravestone: "William Yeats is laid to rest." As a consoling section, this one is interesting because it emphasizes the "nightmare of the dark" in which Europe is lying, "sequestered in its hate." By "the dogs of Europe," Auden is referring to the warring nations who, at the time of writing (1940) were launching into World War Two. Where Auden finds consolation is in the plea that the "poet" will still "with your unconstraining voice" persuade us that it is still possible to "rejoice." Though Yeats may be dead, and "human unsuccess" rife, Auden prays that the "healing fountain" will start up in the current state of chaos, and memory of Yeats's poetry will "teach the free man how to praise."
I think I've covered all the lines you've asked about specifically, although my copy of the poem isn't numbered. Hopefully this gives you a good starting point.

int sin^3 (2theta) sqrt(cos(2theta)) d theta Find the indefinite integral

 
Given to solve,
int sin^3 (2 theta) sqrt(cos(2 theta)) d(theta)
let x= theta (just for convinence)
so,
int sin^3 (2 theta) sqrt(cos(2 theta)) d(theta)
=int sin^3 (2x) sqrt(cos(2x)) dx
 
let 2x= u so , du = 2dx then ,
int sin^3 (2x) sqrt(cos(2x)) dx
=int sin^3 (u) sqrt(cos(u)) (du)/2
=(1/2)int sin^2 (u) sin u sqrt(cos(u)) du
= (1/2)int (1-cos^2 (u)) sin u sqrt(cos(u)) du
 
let cos u =t, so , dt = -sin(u) du
then,
(1/2)int (1-cos^2 (u)) sin u sqrt(cos(u)) du
= (1/2)int (1-t^2) sqrt(t) sin u du
=(1/2)int (1-t^2) sqrt(t) (-dt)
= (-1/2)int (1-t^2) sqrt(t) (dt)
= (-1/2) int (t^(1/2) - t^(5/2))dt
= (-1/2) [(t^(3/2))/(3/2) - t^((5/2)+1)/((5/2)+1)]
= (-1/2) [(t^(3/2))/(3/2) - (t^(7/2))/(7/2)]
but t= cos u = cos(2x) so,
= (-1/2) [((cos(2x))^(3/2))/(3/2) - ((cos(2x))^(7/2))/(7/2)]
= (1/2)[((cos(2x))^(7/2))/(7/2) -((cos(2x))^(3/2))/(3/2)]
but x= theta, so
= (1/2)[((cos(2(theta)))^(7/2))/(7/2) -((cos(2(theta)))^(3/2))/(3/2)]
so,
int sin^3 (2 theta) sqrt(cos(2 theta)) d(theta)
=(1/2)[((cos(2(theta)))^(7/2))/(7/2) -((cos(2(theta)))^(3/2))/(3/2)]
 

sum_(n=0)^oo x^(2n)/((2n)!) Find the radius of convergence of the power series.

sum_(n=0)^oo x^(2n)/((2n)!)
To find the radius of convergence of a series sum a_n , apply the Ratio Test.
L = lim_(n->oo) |a_(n+1)/a_n|
L=lim_(n->oo) | (x^(2(n+1))/((2(n+1))!))/((x^(2n))/((2n)!))|
L=lim_(n->oo) | x^(2(n+1))/((2(n+1))!) * ((2n)!)/x^(2n)|
L= lim_(n->oo)| x^(2n+2)/((2n+2)!) * ((2n)!)/x^(2n)|
L= lim_(n->oo)| x^(2n+2)/((2n+2)*(2n+1)*(2n)!) * ((2n)!)/x^(2n)|
L=lim_(n->oo) |x^2/((2n+2)(2n+1))|
L=|x^2| lim_(n->oo)|1/((2n+2)(2n+1))|
L=|x^2| * 0
L =0
Take note that in Ratio Test, the series converges when L < 1.
Since the value of L is zero, which is less than 1, then the series converges for all values of x.
Therefore, the radius of convergence of the given series is R =oo .

Shakespeare's othello has been described as abundant with racial attitudes. Do you agree

Yes. There is no doubt that Brabantio is initially opposed to his daughter's marriage to Othello because of Othello's race. And Iago makes crude racial remarks and jokes which, arguably, are a sign that his hatred of Othello and his wish to bring Othello down are based at least in part on race hatred.
Othello reveals an insecurity which is probably based to a degree on his awareness of his outsider status, his position as "the Other" among Europeans. This is one reason he is so easily tricked into believing Iago's lies. Another reason is that Othello is more honest and virtuous than the other characters, apart from Desdemona. Like most decent-hearted people, he reflexively finds it difficult to believe others would tell lies and deliberately cause harm, so he fails to see through Iago. One can view Shakespeare, at least in this play, as quite progressive in thought, in his having shown a man of a different race to be superior to Europeans.

College Algebra, Chapter 3, 3.1, Section 3.1, Problem 42

Given the function $\displaystyle f(x) = x^3 $. Find $f(a)$, $f(a+h)$ and the difference quotient $\displaystyle \frac{f(a+h) - f(a)}{h}$ where $h \neq 0$

For $f(a)$
$\displaystyle f(a) = a^3 $ Replace $x$ by $a$

For $f(a+h)$

$
\begin{equation}
\begin{aligned}
f(a+h) &= (a+h)^3 && \text{Replace } x \text{ by } (a + h)\\
\\
&= a^3 + 3a^2h + 3ah^2 + h^3
\end{aligned}
\end{equation}
$


For $\displaystyle \frac{f(a+h)-f(a)}{h}$

$
\begin{equation}
\begin{aligned}
\frac{f(a-h)-f(a)}{h} &= \frac{a^3 + 3a^2h + 3ah^2 + h^3 - a^3}{h} && \text{Substitute } f(a+h) = a^3 + 3a^h + 3ah^2 + h^3 \text{ and } f(a) = a^3\\
\\
&= \frac{3a^2h + 3ah^2 + h^3}{h} && \text{Simplify}\\
\\
&= \frac{\cancel{h}\left( 3a^2 + 3ah + h^2 \right)}{\cancel{h}} && \text{Cancel out like terms}\\
\\
&= 3a^2 + 3ah + h^2
\end{aligned}
\end{equation}
$

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 13

To determine the convergence or divergence of a series sum a_n using Root test, we evaluate a limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
We may apply the Root Test to determine the convergence or divergence of the series sum_(n=1)^oo 1/5^n.
For the given series sum_(n=1)^oo 1/5^n, we have a_n =1/5^n.
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |1/5^n|^(1/n)=lim_(n-gtoo) (1/5^n)^(1/n)
Apply the Law of Exponents: (x/y)^n =(x^n/y^n) .
lim_(n-gtoo) (1/5^n)^(1/n)=lim_(n-gtoo) 1^(1/n)/5^(n*(1/n))
=lim_(n-gtoo) 1^(1/n)/5^(n/n)
=lim_(n-gtoo) 1^(1/n)/5^1
=lim_(n-gtoo) 1^(1/n)/5
Evaluate the limit.
lim_(n-gtoo) 1^(1/n)/5 =1/5 lim_(n-gtoo) 1^(1/n)
= 1/5*1^(1/oo)
= 1/5 * 1^0
= 1/5*1
=1/5
The limit value L =1/5 satisfies the condition: Llt1 .
Thus, the series sum_(n=1)^oo 1/5^n is absolutely convergent.

Which incident impresses you the most from the play Romeo and Juliet?

Personally, the incident that impresses me the most from this play is when Juliet takes her life into her own hands—literally and figuratively—and decides to fake her own death, deceive her family and her nurse, and make one last-ditch effort to be with her beloved, Romeo. She is so incredibly young, just thirteen, and she has so many reasonable fears about taking this step, but she bravely faces those fears and goes for it. Juliet worries the potion Friar Lawrence gave her might not work at all—then what will she do? She worries she could wake up earlier than planned and find herself trapped in her family's vault. She also has concerns that the mixture might kill her. Juliet must act alone, without anyone's support, taking a really dramatic step, and she does so courageously. I find this extremely impressive.

y = sqrt(9-x^2) Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

To find the volume of a solid by revolving the graph of y =sqrt(9-x^2) about the x-axis, we consider the bounded region in between the graph and the x-axis. To evaluate this, we apply the Disk method by using a rectangular strip  perpendicular to the axis of rotation. As shown on the attached image, we consider a vertical rectangular strip with a thickness =dx. 
 
We follow the formula for the Disk Method in a form of: V = int_a^b pir^2 dx or V = pi int_a^b r^2 dx  where r is the length of the rectangular strip.
 In this problem, we let the length of the rectangular strip=y_(above)-y_(below) .
 Then r =sqrt(9-x^2) -0=sqrt(9-x^2)
Boundary values of x: a= -3 to b=3 .
Plug-in the values on the formula V = pi int_a^b r^2 dx , we get:
V =pi int_(-3)^3 (sqrt(9-x^2))^2 dx
V =pi int_(-3)^3 (9-x^2) dx
Apply basic integration property: int (u-v)dx = int (u)dx-int (v)dx .
V =pi *[ int_(-3)^3 (9) dx- int_(-3)^3(x^2) dx]
For the integral of int_(-3)^3 (9) dx , we apply basic integration property: int c dx = cx .
int_(-3)^3 (9) dx =9x|_(-3)^3
For the integral of  int_(-3)^3(x^2) dx , we apply Power rule for integration: int x^n dx = x^(n+1)/(n+1) .
int_(-3)^3(x^2) dx = x^((2+1))/((2+1))|_(-3)^3 .
                    =x^3/3|_(-3)^3.
Then,
V =pi *[ int_(-3)^3 (9) dx-int_(-3)^3(x^2) dx]
V =pi *[ 9x-x^3/3]|_(-3)^3
Apply definite integration formula: int_a^b f(y) dy= F(b)-F(a) .
V =pi *[ 9(3)-(3)^3/3] -pi *[ 9(-3)-(-3)^3/3]
V =pi *[ 27-27/3] -pi *[ -27- (-27)/3]
V =pi *[ 81/3-27/3] -pi *[ (-81)/3- (-27)/3]
V =pi *[ 54/3] -pi *[ (-54)/3]
V =(54pi)/3 - ((-54pi)/3)
V =(54pi)/3 + (54pi)/3
V =(108pi)/3
V =36pi or 113.1 (approximated value)

What are 3 of the most significant examples of courage demonstrated in To Kill a Mockingbird?

Three significant examples of courage demonstrated in To Kill a Mockingbird are (1) the stand-off at the jailhouse with Atticus and the Old Sarum Bunch, (2) the appeal by Atticus to the consciences of the jurors as he exhorts them to follow the Constitution and their consciences, and (3) Arthur Radley's heroic defense of Jem when he is attacked by Bob Ewell.
1. Jailhouse stand-off
When a number of cars with the Old Sarum Bunch pull up, Atticus, who sits in an office chair in front of the jail where Tom Robinson is held, calmly folds his newspaper and stands up. Mr. Walter Cunningham tells Atticus, "You know what we want." Atticus suggests that they leave, saying, "Heck Tate's around somewhere"; however, one of the men contradicts Atticus: "Heck's bunch's so deep in the woods they won't get out till mornin," implying that the sheriff has falsely been called away.Although his hands shake when he hears that the sheriff is gone, Atticus's voice remains calm. "Well then,...that changes things." But, Atticus does not move out of the mob's way.
Fortunately, Jem and Scout, who have come to find their father, refuse to go home and after Scout speaks to Mr. Cunningham with such words that remind Mr. Cunningham of the kind acts of Atticus toward him, he tells his men, "Let's clear out....Let's get going, boys."
2. Atticus's appeal to the jurors to follow their consciences and the Constitution
After all the vilifications about Atticus by citizens in the town, as well as his acquaintance with the conventions of the townspeople, it is a courageous man who makes an appeal in court to the jury before the citizens of Maycomb that contradicts their beliefs and culture. Reminding the jury of the American courts' being the "great levelers" in which "all men are created equal," Atticus addresses them:

"Gentlemen, a court is no better than each man of you sitting before me on this jury....I am confident that you...will review without passion the evidence you have heard, come to a decision, and restore this defendant to his family. In the name of God, do your duty."

3. Arthur Radley's act of bravery
When Boo Radley hears what goes on outside his window, he apparently investigates what has caused the Finch children to sound distressed. Without hesitation, then, he rushes outside, wrestles with their attacker, Bob Ewell, and the vicious man is stabbed by his own knife. After saving the children's lives, Boo carries the children home. It is a grateful Atticus Finch who says, "Thank you for my children, Arthur."