# Thread: Volume Problem - "Shell" Method

1. ## Volume Problem - "Shell" Method

Hi again friends,

I would appreciate help on 2 more problems please.

1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating about the y-axis and the region R bounded by the curve y=2x^2-x^3 (two ex squared minus ex cubed) and the x-axis.

2. Use the method of cylindrical shells to find the volume of the solid obtained rotating the region R by the curves y = x^3 (ex cubed), y = 0 and x = 2 and about the line x = 3

2. Hello, nmq3b!

Can you sketch the region, or do you have a graphing calculator?
A good sketch is absolutely essential.

1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating
about the y-axis, the region R bounded by the curve $y\:=\:2x^2-x^3$ and the x-axis.
Code:
            |
*    |        *
|     *:::::*
*   |    *:::::::*
*  |  *::::::::::
- - - * - - - - - - * - -
|             2
|              *
|               *
Shells Formula: . $V \;=\;2\pi\int^b_a\text{(radius)(height)}\,dx$

In this problem: . $\text{radius} \:=\:x,\;\text{height} \:=\:2x^2-x^3$

Hence: . $V \;=\;2\pi\int^2_0x(2x^2-x^3)\,dx$

I assume you can finish it . . .

2. Use the method of cylindrical shells to find the volume of the solid obtained rotating
the region R boounded by the curves $y\:=\:x^3,\;y = 0,\;x = 2$ about the line $x = 3$
Code:
            |       *   :
|       |   :
|      *|   :
|     *:|   :
|  .*:::|   :
- - - - * - - - + - + - -
*   |       2   3
*     |
*      |

This time we have: . $\text{radius } = 3-x,\;\text{height} = x^3$

Hence: . $V \;=\;2\pi\int^2_0(3-x)x^3\,dx$