For the region bounded by y=2 and y =4-x^2/4 revolved about the x-axis, we may apply Washer method for the integral application for the volume of a solid.
As shown on the attached image, we are using vertical rectangular strip that is perpendicular to the x-axis (axis of revolution) with a thickness of "dx" . In line with this, we will consider the formula for the Washer Method as:
V = pi int_a^b [(f(x))^2-(g(x))^2]dx
where f(x) as function of the outer radius, R
g(x) as a function of the inner radius, r
For each radius, we follow the y_(above) - y_(below) , we have y_(below)=0 since it a distance between the axis of rotation and each boundary graph.
For the inner radius, we have: g(x) =2-0=2
For the outer radius, we have: f(x) =(4-x^2/4 )-0=4-x^2/4
To determine the boundary values of x, we equate the two values of y's:
4-x^2/4 =2
-x^2/4 =2-4
-x^2/4 =-2
(-4)(-x^2/4 ) =(-4)(-2)
x^2=8 then x= +-sqrt(8) or +2sqrt(2) and -2sqrt(2)
Then, boundary values of x: a=-2sqrt(2) and b=2sqrt(2) .
Plug-in the values in the formula V = pi int_a^b( (f(x))^2 -(g(x))^2) dx , we get:
V =pi int_(-2sqrt(2))^(2sqrt(2)) [(4-x^2/4)^2 -2^2]dx .
Expand using the FOIL method on: (4-x^2/4)^2 = (4-x^2/4)(4-x^2/4)= 16-2x^2+x^4/16 and 2^2=4 .
The integral becomes:
V =pi int_(-2sqrt(2))^(2sqrt(2)) [16-2x^2+x^4/16 -4]dx
V =pi int_(-2sqrt(2))^(2sqrt(2)) [12-2x^2+x^4/16 ]dx
Apply basic integration property: int (u+-v+-w)dx = int (u)dx+-int (v)dx+-int(w)dx to be able to integrate them separately using Power rule for integration: int x^n dx = x^(n+1)/(n+1) .
V =pi *[int_(-2sqrt(2))^(2sqrt(2))(12) dx -int_(-2sqrt(2))^(2sqrt(2)) (2x^2) dx + int_(-2sqrt(2))^(2sqrt(2)) (x^4/16)dx]
V =pi *[12x-2 *x^3/3+1/16*x^5/5 ]|_(-2sqrt(2))^(2sqrt(2))
V =pi *[12x-(2x^3)/3+x^5/80 ]|_(-2sqrt(2))^(2sqrt(2))
Apply the definite integral formula: int _a^b f(x) dx = F(b) - F(a) .
V =pi *[12(2sqrt(2))-(2(2sqrt(2))^3)/3+(2sqrt(2))^5/80 ]-pi *[12(-2sqrt(2))-(2(-2sqrt(2))^3)/3+(-2sqrt(2))^5/80 ]
V =pi *[24sqrt(2)-(32sqrt(2))/3+(8sqrt(2))/5 ] -pi *[-24sqrt(2)+(32sqrt(2))/3-(8sqrt(2))/5 ]
V =(224sqrt(2)pi)/15 -(-224sqrt(2)pi)/15
V =(224sqrt(2)pi)/15 +(224sqrt(2)pi)/15
V =(448sqrt(2)pi)/15 or 132.69 (approximated value)
Wednesday, May 15, 2013
y = 2 , y = 4-x^2/4 Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.
Subscribe to:
Post Comments (Atom)
Why is the fact that the Americans are helping the Russians important?
In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...
-
There are a plethora of rules that Jonas and the other citizens must follow. Again, page numbers will vary given the edition of the book tha...
-
The only example of simile in "The Lottery"—and a particularly weak one at that—is when Mrs. Hutchinson taps Mrs. Delacroix on the...
-
A good thesis statement presents a claim (an interpretive stance on a story that can be defended using textual evidence) and is a position w...
-
The given two points of the exponential function are (2,24) and (3,144). To determine the exponential function y=ab^x plug-in the given x an...
-
What does the hot air balloon symbolize? To the Assad son who buys the hot air balloon, it symbolizes a kind of whimsy that he can afford. B...
-
The play Duchess of Malfi is named after the character and real life historical tragic figure of Duchess of Malfi who was the regent of the ...
-
Allie’s baseball mitt is extremely important to Holden in The Catcher in the Rye. It is a symbol of Allie since it was important to his brot...
No comments:
Post a Comment