I'm working on integration and change of variables. I'm not sure about how to do an exercise by using spherical and cylindrical coordinates. I think I can understand the process just seeing how it works for the spherical ones.
Calculate the volume of the set between the cone $y^2\sin^2a=(x^2+z^2)\cos^2a$ and the sphere $x^2+y^2+z^2=b^2$.
First, I've replaced $x=r\cdot \sin\theta\cdot \cos\phi$, $y=r\cdot \sin\theta\cdot \sin\phi$ and $z=r\cdot \cos\theta$ in both expressions and I've obtained: $0=-r^2\cdot \sin^2\theta\cdot \sin^2\phi-r^2\cdot \sin^2a$ for the cone and $r=b$ for the sphere.
Then, I thought of finding the intersection between the cone and the sphere: $y^2=b^2-x^2-z^2\Longrightarrow(b^2-x^2-x^2)\sin^2a=(x^2+z^2)\cos^2a$.
By using $\sin^2a+\cos^2a=1$, I've got: $b^2\sin^2a=x^2+z^2\Longrightarrow b^2\sin^2a=-b^2\sin^2a=-b^2\sin^2\theta \sin^2\phi-b^2$
I don't know how to go on to obtain the limits for $\theta$ and $\phi$ since I think I have the value of $r$ ($r=b$), and I don't know if it's ok what I've done neither.
Thanks!