Using Divergence Theorem to evaluate the integral $ \iint_{\Sigma} \langle x, y^3, -z\rangle. d\vec{S}$

Question

Find the value of $$ \iint_{\Sigma} \langle x, y^3, -z\rangle. d\vec{S} $$ where $ \Sigma $ is the sphere $ x^2 + y^2 + z^2 = 1 $ oriented outward by using the divergence theorem.

So I calculate $\operatorname{div}\vec{F} = 3y^2 $ and then I convert $ x, y, z $ into $ x = p\sin \phi \cos \theta, y = p\sin \phi \sin \theta, z = p\cos \phi $ but then I got stuck from that point.

Answer 1

You therefore want the volume integral

$$\iiint_V 3y^2 \ dV = \int_0^1 \int_0^{2\pi} \int_0^\pi 3(p\sin\phi \sin\theta)^2 . p^2 \sin\phi \ dp \ d\theta \ d\phi = \ \cdots$$

Since the volume of integration is a rectangular box in $(p,\theta, \phi)$ coordinates, you can separate out the integrating variables to make it easy to evaluate the overall integral.

Answer 2

$$\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^1 \left(\rho^2 \sin \phi \right)\underbrace{(\sqrt{3}\rho \sin \phi \sin \theta)^2}_{3y^2 \ \textrm{in spherical}} \ d\rho \ d\phi \ d\theta$$

Answer 3

You need to evaluate: $$\int_0^1 3 p^4 \ dp\int_0^{2\pi} \sin^2\theta \ d\theta \int_0^\pi \sin^3\phi \ d\phi $$