So I need to find the volume of G using spherical coordinates. My problem is in finding the new boundaries. I know that G is a sphere with a cilinder inside it removed. What I don't know, is how I can get the boundaries for r, $\phi$ and $\theta$. I thought that $r$ had to be between $\frac{1}{sin(\theta)}$ and 2 but that was incorrect. Can someone explain to me why that value for $r$ is wrong and if there is a good way of attacking such problems?
Thanks for reading,
K.
The point is to find the intersection of the two surfaces:
$x^2+y^2=1$ and $x^2+y^2+z^2=4$, are in spherical coordinates, $r_0^2\sin^2\theta_0=1$ and $r_0^2=4$, respectively; so $\sin\theta_0=1/2$
The limits are $0\leq\phi\leq 2\pi$, $\pi/6\leq\theta\leq\pi-\pi/6$ and $1/\sin\theta\leq r\leq 2$
The volume is:
$$V=\int_0^{2\pi}d\phi\int_{\pi/6}^{5\pi/6}\int_{1/\sin\theta}^{2}r^2\sin\theta drd\theta$$
Added
This is the section along the plane $yz$, the lines being for the cylinder and the circle for the sphere.