
We integrate over regions in spherical coordinates.

Another way to generalize polar coordinates to three dimensions is with spherical coordinates.
Consider the point $(\rho ,\theta ,\phi )=(2,-\pi /4,\pi /4)$ in spherical coordinates. What is this point when expressed in $(x,y,z)$-coordinates?

### Triple integrals in spherical coordinates

If you want to evaluate this integral you have to change $R$ to a region defined in $(x,y,z)$-coordinates, and change $\d V$ to some combination of $\d x\d y\d z$ leaving you with some iterated integral: Now consider representing a region $R$ in spherical coordinates and let’s express $\d V$ in terms of $\d \rho$, $\d \phi$, and $\d \theta$. To do this, consider the diagram below:

Here we see

We may now state at theorem:

Write down a triple integral in spherical coordinates that will compute the volume of a sphere of radius $a$.