I'm currently trying to understand cylindrical and spherical coordinates more. I understand them by rote and less by actually understanding what they mean/why we should use them.
Right now, I'm working on the following problem:
There is a meridional zone of height $h=c-b$ that lies on the sphere $r^2+z^2=a^2$ between the planes $z=b$ and $z=c$, where $0<b<c<a$. Apply Eq.(1) to show that the area of this zone is $A=2\pi ah.$
This is what I have done so far:
$ z = \sqrt{a^2-r^2}, z= b, z=c$
$\int_0^{2\pi}\int_0^a \sqrt{r^2+r^2*(\frac{-r}{\sqrt{a^2-r^2}})^2} dr d\theta$
$\int_0^{2\pi}\int_0^a r\sqrt{1+\frac{r^2}{a^2-r^2}} dr d\theta$
$\int_0^{2\pi}\int_0^a r\sqrt{\frac{1}{a^2-r^2}} dr d\theta$
I'm not sure if this was right at all. I don't know if I'm setting up right - because now I have no idea how to do this integral.
Thanks for any help!