I have the following triple integral in spherical coordinates $(r,\theta,\phi)$: $$\int_0^{2\pi}\int_0^\pi\int_0^RCr^3\hat\theta\cdot r^2dr\sin{\theta}d\theta d\phi$$
How do I handle the $\hat\theta$? If I ignore it, I get $\frac{2}{3}\pi CR^6$. So is my answer the vector $\frac{2}{3}\pi CR^6\hat\theta$? Do I need to integrate the unit vector $\hat\theta$? If so, how?
Best Answer
Yes, you need to integrate this, as $\hat \theta$ is a function of $\theta, \phi$. A straightforward way to attack the problem is to write $\hat \theta$ as a linear combination of $\hat x, \hat y, \hat z$, but with the components expressed in terms of $r, \theta, \phi$.