Hello, mollymcf2009!

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

Use the method of cylindrical shells to find the volume of the solid

obtained by rotating the region about the $\displaystyle x$-axis.

We have a parabola: .$\displaystyle x \:=\:(y-3)^2 + 8$

. . The vertex is $\displaystyle (8,3)$ and it opens to the right.

$\displaystyle x = 9$ is a vertical line.

The graph looks like this: Code:

| | *
| *
| *:::|
| *::::::|
| *::::::::|
| (8,3)*:::::::::|
| *::::::::|
| *::::::|
| *:::|
| *
| | *
- - + - - - - - - - - - + - - - - - - -
| 9

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

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

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

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