Help needed with a Calculus/Vector question

Question

this is my question, any help would be appreciated.

Consider the vector $D= 3 \sin(\theta)$ ar (unit vector).

Evaluate the integral $$\iint_S D ~dS,$$ where $S$ is the surface of a sphere with radius $r=5$ centered at the origin.

Answer 1

In terms of polar coordinates, the surface area element at constant $r$, in your case $r=5$, is $r^2\sin\theta d\theta d\phi$. Let $\hat{n}$ denote the unit vector parallel to, $D$ so your integral is$$\int_0^{2\pi}d\phi\int_0^\pi d\theta r^2\sin\theta D\cdot\hat{n}=150\pi\int_0^\pi\sin^2\theta d\theta=75\pi^2.$$