For the region bounded by y=3-x , y=0 , y=2 , and x=0 revolved about the line x=5 , we may apply Washer method for the integral application for the volume of a solid.
The formula for the Washer Method is:
V = pi int_a^b [(f(x))^2-(g(x))^2]dx
or
V = pi int_a^b [(f(y))^2-(g(y))^2]dy
where f as function of the outer radius
g as a function of the inner radius
To determine which form we use, we consider the horizontal rectangular strip representation that is perpendicular to the axis of rotation as shown on the attached image. The given strip has a thickness of "dy " which is our clue to use the formula:
V = pi int_a^b [(f(y))^2-(g(y))^2]dy
For each radius, we follow the x_2-x_1 . We have x_2=5 since it a distance between the axis of rotation and each boundary graph.
For the inner radius, we have: g(y) = 5-(3-y) simplified to g(y)=2+y
Note: y_(below) for the inner radius is based from y =3-x rearrange into x= 3-y
For the outer radius, we have: f(y) = 5-0 simplified to f(y)=5 .
Then the boundary values of y is a=0 and b =2 .
Then the integral will be:
V = pi int_0^2 [(5)^2-(2+y)^2]dy
Expand using the FOIL method on:
(2+y)^2 = (2+y)(2+y)= 4+4y+y^2 and 5^2=25 .
The integral becomes:
V = pi int_0^2 [25 -(4+4y+y^2 )]dy
Simplify:
V = pi int_0^2 [25 -4-4y-y^2 ]dy
V = pi int_0^2 [21-4y-y^2 ]dy
Apply basic integration property:
int (u-v-w)dy = int (u)dy-int (v)dy-int (w)dy
V = pi [ int_0^2 21dy - int_0^2 4ydy -int_0^2 y^2dy]
For the integration of int 21 dy , we apply basic integration property: int c dx = cx .
For the integration of int_0^2 4ydy and int_0^2 y^2dy , we apply the Power rule for integration: int y^n dx = y^(n+1)/(n+1) .
V = pi [ 21y - 4*y^2/2 -y^3/3]|_0^2
V = pi [ 21y - 2y^2 -y^3/3]|_0^2
Apply the definite integral formula: int _a^b f(x) dx = F(b) - F(a) .
V = pi [ 21(2) - 2(2)^2 -(2)^3/3]-pi [ 21(0) - 2(0)^2 -(0)^3/3]
V = pi [ 42 - 8 -8/3]-pi [ 0-0-0]
V = pi [ 94/3]-pi[0]
V =(94pi)/3 or 98.44 (approximated value)
Sunday, October 15, 2017
Calculus of a Single Variable, Chapter 7, 7.2, Section 7.2, Problem 20
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