Whenever I have been presented with integrals as (*), I have always used some sort of symmetry to get around actually calculating the integral. Now it just struck me that I have no idea how to "mechanically" calculate (*):
$$
\iiint \left(f(r)\hat{r}\right)\left(r^2\sin\phi\right) drd\phi d\theta \qquad \textbf{(*)}
$$
If it is necessary to define what region you are talking about for the question to make sense, I'll pick the unit sphere.
Would this have been the integral
$$
\int f(x)\hat{x} dx
$$
I would have proceeded by moving the unit vector outside of the integral and calculated as usual
$$
\hat{x}\int f(x)\cdot dx = \hat{x}\left(F(x) +C\right)
$$
However, the problem with the integral is that the direction of $\hat{r}$ changes as $\phi$ and $\theta$ changes. Therefore I don't see how I could treat the unit vector as a constant and have no idea how to start with (*).
I apologize in advance if this turns out to be a duplicate. How to integrate a vector function in spherical coordinates? verifies my gut feeling that $\hat{r}$ is not constant but doesn't make things much clearer for me than that.
Best Answer
Since $\hat{\mathbf{r}} = \sin\theta\cos\phi\,\hat{\mathbf{x}} + \sin\theta\sin\phi\,\hat{\mathbf{y}} + \cos\theta\,\hat{\mathbf{z}}$, you can write
$$ \iiint f(r)\,\hat{\mathbf{r}}\,r^2\sin\theta\,dr\,d\phi\,d\theta = \iiint f(r) \left(\sin\theta\cos\phi\,\hat{\mathbf{x}} + \sin\theta\sin\phi\,\hat{\mathbf{y}} + \cos\theta\,\hat{\mathbf{z}}\right)r^2\sin\theta\,dr\,d\phi\,d\theta $$ which is the sum of three separate integrals: $$ \hat{\mathbf{x}} \iiint f(r)\sin\theta\cos\phi\,r^2\sin\theta\,dr\,d\phi\,d\theta = \hat{\mathbf{x}} \int f(r)\,r^2\,dr \int\sin^2\theta\,d\theta \int\cos\phi\,d\phi $$ $$ \hat{\mathbf{y}}\iiint f(r)\,\sin\theta\sin\phi\,r^2\sin\theta\,dr\,d\phi\,d\theta = \hat{\mathbf{y}} \int f(r)\,r^2\,dr \int\sin^2\theta\,d\theta \int\sin\phi\,d\phi $$ $$ \hat{\mathbf{z}}\iiint f(r)\,\cos\theta\,r^2\sin\theta\,dr\,d\phi\,d\theta = \hat{\mathbf{z}} \int f(r)\,r^2\,dr \int\sin\theta\cos\theta\,d\theta \int d\phi $$
where $\hat{\mathbf{r}}$ is the unit vector along the radial direction but $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ are the Cartesian unit vectors.
(Note that I'm using the convention where the volume element is $r^2\sin\theta\,dr\,d\phi\,d\theta\,$.)