# Shell method Volume of a solid problem

• Nov 30th 2008, 06:44 PM
calnix
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 https://webwork.math.lsu.edu/webwork...394a860011.png, https://webwork.math.lsu.edu/webwork...16fea26871.png and https://webwork.math.lsu.edu/webwork...3084628a01.png 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>
• Nov 30th 2008, 07:44 PM
Soroban
Hello, calnix!

Did you make a sketch?

Quote:

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

Code:

                  |       *          *    |    *             *    |    *|       *  *      |    |:*  *         *        |    |:::*       * *        |    |::*:*           *      |    |*  :       *      *  |  * |  : *     - - - - - - - * - - + - + - -                   |    1½  2                   |

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

The two parabolas intersect at $\displaystyle x = \pm2$

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

Go for it!

• Nov 30th 2008, 08:08 PM
calnix
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 https://webwork.math.lsu.edu/webwork...635d2bdfc1.png for https://webwork.math.lsu.edu/webwork...23102a6ff1.png 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?