How do I use Gauss' theorem to evaluate a surface integral?

[epyfathom]

Find and evaluate numerically

x^10 + y^10 + z^10 dSx^2 + y^2 + z^2 =4It says you're supposed to use gauss' divergence thm to convert surface integral to volume integral, then integrate volume integral by converting to spherical coordinates... I can do the second part but how do i use gauss' thm...? my prof was really bad at explaining this.Thanks.

 

[HallsofIvy]

Well, I would think that the first thing you would do is look up "Gauss' theorem" (perhaps better known as the "divergence theorem"). According to Wikipedia, Gauss' theorem says that
\int\int\int (\nabla\cdot \vec{F}) dV= \oint\int \vec{F}\cdot\vec{n}dS
where \vec{n} is the normal vector to the surface at each point.

Here, you are not given a vector function but, fortunately, Wikipedia also notes that "Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven:
\int\int\int \nabla f dV= \oint\int f dS"

So, since you are asked to use Gauss' theorem to evaluate a surface integral, you are intended to find \nabla f and integrate that over the region- the ball of radius 2.

Then- first step- what is \nabla (x^{10}+ y^{10}+ z^{10})?