Hi! I am studying for an exam and working on understanding spherical coordinate integrals. For the integral below there is a cone and a sphere. I saw a solution to this problem which involved translating to spherical coordinates to get a triple integral. The integral solved was(p^2)*sin(phi). How does one go about obtaining this? I understand how to calculate the bounds. Thanks!
$\rho^2\sin\phi$ is the "fudge factor" in spherical coordinates; i.e., $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.
the Jacobian.
$dx \ dy \ dz = \left|\det\begin{bmatrix} \frac {\partial x}{\partial \rho} &\frac {\partial x}{\partial \theta}&\frac {\partial x}{\partial \phi}\\ \frac {\partial y}{\partial \rho} &\frac {\partial y}{\partial \theta}&\frac {\partial y}{\partial \phi}\\ \frac {\partial z}{\partial \rho} &\frac {\partial z}{\partial \theta}&\frac {\partial z}{\partial \phi}\end{bmatrix}\right| \ d\rho\ d\phi \ d\theta = \rho^2\sin\phi \ d\rho\ d\phi \ d\theta$
What does it mean? Starting at some $(\rho, \phi, \theta)$, nudge each of the variables. $(\rho + d\rho,\phi, \theta), (\rho,\phi+d\phi, \theta)$, etc. Nudging each of these, creates a "box" of some volume. What is the volume?