What is the volume of the following region, described in spherical coordinates?
1 <= ρ <= 9, 0 <= ϴ <= π/2, π/6 <= ϕ <= π/4
Attempt:
Since ρ^2 = x^2 + y^2 + z^2, the first part appears to describe the surface between two concentric spheres, centered at the origin with radii of 1 and 0 respectively.
ϴ=0 is a half xz plane and ϴ=π/2 is a half yz plane.
I know that ϕ has to do with the angular range with respect to the z axis, but I am not sure how to restrict these values and determine an integral of which I can integrate to find the volume.
Thanks!
From your description it sounds like $\theta$ is the azimuthal angle (ranges $0$ to $2\pi$) and $\phi$ is the polar angle (ranges $0$ to $\pi$). In that case the spherical volume element is $\rho^2\sin\phi d\rho d\phi d\theta$. The problem statement just gives you the range to integrate over. So you have $$ V = \int_1^9d\rho\int_{\pi/6}^{\pi/4}d\phi\int_{0}^{\pi/2}d\theta \rho^2\sin\phi.$$