# Thread: Shell method Volume of a solid problem

1. ## Shell method Volume of a solid problem

Here's the problem:

Use the Shell Method to compute the volume of the solids obtained by rotating the region enclosed by the graphs of the functions , and about the y-axis.

I can solve problems when I am given one or two functions, an interval, and an axis to rotate about...but I am completely lost on how to solve this kind of problem. Any help would be great, thanks>

2. Hello, calnix!

Did you make a sketch?

Use the Shell Method to compute the volume of the solid obtained
by rotating the region bounded by: $y \:=\:x^2,\;y \:=\:8-x^2,\;x \:=\: \tfrac{3}{2}$ about the y-axis.
Code:
                  |
*           *     |     *
*    |    *|
*  *       |     |:*  *
*         |     |:::*
* *        |     |::*:*
*      |     |*  :
*       *   |   * |   : *
- - - - - - - * - - + - + - -
|    1½   2
|

Formula: . $V \;=\;2\pi \int^b_a x\left(y_2 - y_1\right)\,dx$

The two parabolas intersect at $x = \pm2$

We have: . $V \;=\;2\pi\int^2_{\frac{3}{2}} x\left[(8-x^2) - x^2\right]\,dx$

Go for it!

3. Thanks a lot! I didn't know I'd use the given x-value and then find the other for the bounds. Appreciate it a lot! If you don't mind, I have another problem:

Use both the Shell and Disk Methods to calculate the volume of the solid obtained by rotating the region under the graph of for about:
the x-axis:____
the y_axis:____

I need to find what values of the axis the region is rotated around. Would I just use the 0 and 13/3 for the bounds?