I cannot figure out how to solve this. The region is bounded by $xy=7$, $y=1$, $y=8$ and $x=9$. Revolving about the line $x=9$.
I came up with the following integral to solve this:
$\pi\int_{1}^{8}(9)^2-(7/y)^2dy$
The answer I got is $1646.587$.
What am I doing wrong? Did I set up the problem incorrectly?
A general tip in the case you are not revolving around a coordinate axis is to transform your expressions so that you are. Your situation looks like this:
If you shift $9$ steps to the left (i.e. replace all $x$ by $x+9$), then it looks like this
So, the domain that revolves is bounded by the curves $$ (x+9)y=7,\quad y=1,\quad y=8,\quad\text{and}\quad x=0. $$ The first equation gives $$ x=\frac{7}{y}-9. $$ Thus, the volume you look for is given by the integral $$ \int_1^8\pi(7/y-9)^2\,dy. $$