# Thread: Volumes by Cylindrical Shells

1. ## Volumes by Cylindrical Shells

Hi everyone!

Could someone show me a graph of this so I can verify mine? Also, I can't figure out if I need u-substitution on this or not. I've tried to work it both ways and am not getting the correct answer.

Consider the given curves:

$x = 8 + (y-3)^2$ and $x=9$

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

Thanks!!!

2. Attachment 9801

Your integral should start off looking like $2\pi\int_2^4(9-(8+(y-3)^2))ydy$ and you should end up with $8\pi$.
I would suggest just multiplying out everything and then integrating term by term.

3. Beautiful! Thank you so much!!

4. Hello, mollymcf2009!

Given the region bounded by: . $x \:= \:8 + (y-3)^2\:\text{ and }\:x\,=\,9$

Use the method of cylindrical shells to find the volume of the solid
obtained by rotating the region about the $x$-axis.

We have a parabola: . $x \:=\:(y-3)^2 + 8$
. . The vertex is $(8,3)$ and it opens to the right.
$x = 9$ is a vertical line.

The graph looks like this:
Code:
      |                   |     *
|                   *
|               *:::|
|            *::::::|
|          *::::::::|
|    (8,3)*:::::::::|
|          *::::::::|
|            *::::::|
|               *:::|
|                   *
|                   |     *
- - + - - - - - - - - - + - - - - - - -
|                   9

The curves intersect at $(9,2) \text{ and }(9,4)$

The formula is: . $V \;=\;2\pi \int^b_a y\bigg[x_{\text{right}} - x_{\text{left}}\bigg]\,dy$

So we have: . $V \;=\;2\pi\int^4_2y\bigg(9 - \left[8-(y-3)^2\right]\bigg)\,dy$

. . and we must evaluate: . $V \;=\;2\pi \int^4_2\left(y^2 - 6y + 10\right)\,dy$