multivariable calculus charge density question

Question

The sphere given by $x^{2} + y^{2} + z^{2} = 4$ is submerged in an electric field with charge density given by $f(x, y, z) = x^{2} + y^{2}$. Find the total amount of electric charge on this surface.

I am stuck on what integral formula to use.

Answer 1

We can solve this by integrating over the surface of the sphere in spherical coordinates $$x^2+y^2+z^2=4 \rightarrow r=2$$ $$f(x,y,z)=x^2+y^2 \rightarrow f(\theta,\phi)=sin^2(\theta)$$ $$dA = r^2sin(\theta)d\theta d\phi$$ $$\iint_{S(A)}sin^2(\theta)dA=\int_{0}^{2\pi}\int_{0}^{\pi}4sin^3(\theta)d\theta d\phi= \frac{32\pi}{3}$$