`y = 9-x^2 , y = 0 , x = 2 , x = 3` Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Aside from Shell method,we may apply the Washer method using the formula:

`V =pi int_a^b ( (f(y))^2 -(g(y))^2) dy`

where `f(y)` as function of the outer radius 

        `g(y)` as function of the inner radius

In washer method, the rectangular strip is perpendicular to the axis of rotation.

 For this problem,  we may  let:

Boundary values of y: ` a=0 ` to` b=5` .

Note: the highest value of y for the bounded region is along `x=2` .

 Plug-in `x=2` on `y =9-x^2` , we get `y=9-(2)^2=9-4 =5` .

Inner radius: g(y) = 2-0 = 2

Outer radius: `f(y) = sqrt(9-y) -0 = sqrt(9-y)`

Note: We can rearrange` y = 9-x^2` as `x^2 =9-y.` Take the square root on both side to express it as `x= sqrt(9-y)` 

Then, wee may set-up the integral as:

`V =pi int_0^5 ( (sqrt(9-y))^2 -(2)^2) dy`


`V =pi int_0^5 ( 9-y -4) dy`

`V =pi int_0^5 ( 5-y ) dy`

Apply basic integration property: `int (u-v)dy = int (u)dy-int (v)dy` .

`V =pi*[ int_0^5 ( 5)dy-int_0^5(y ) dy]`

`V = pi * [5y -y^2/2]|_0^5`

Apply definite integration formula: `int_a^b f(y) dy= F(b)-F(a)` .

`V = pi * [5(5) -(5)^2/2]-pi * [5(0)-(0)^2/2]`

`V = pi * [25 -25/2]-pi * [0-0]`

`V = pi * [25/2]-pi * [0]`

`V =(25pi)/2 - 0`

`V =(25pi)/2` or `39.27` (approximated value)

Image (1 of 1)
Approved by eNotes Editorial
An illustration of the letter 'A' in a speech bubbles

Let's use the shell method for finding the volume of the solid.

The volume of the solid (V) generated by revolving about the y-axis the region between the x-axis and the graph of the continuous function`y=f(x), a <= x<= b`  is,

`V=int_a^b2pi(shell radius) (shell height)dx`


Given ,`y=9-x^2 , y=0 , x=2 , x=3`










See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial