# To evaluate using Gauss Divergence Theorem

Using Gauss Divergence Theorem, evaluate the integral $\int_{S}\int F.\hat n dS$

where $F=(4xz,-y^2,4yz)$ . S is surface of solid bounded by sphere $x^2+y^2+z^2=10$ and paraboloid $x^2+y^2=z-2$ and $\hat n$ is outward unit normal

ATTEMPT

Now i used Gauss Divergence theorem and i am having problem setting up triple integral. I tried in cylindrical and spherical coordinates but solving integral becomes a total mess. My textbook has evaluated line integral along bounding curve $x^2+y^2=1$ and $z=3$ which i donot seem to understand why?

Thanks

By the divergence theorem, your integral equals $$\iiint_E4z+2y\; dV,$$ where $E$ is the region bounded by the sphere and the paraboloid, i.e., in cylindrical coordinates $$E=\{(r,\theta,z)|0\le \theta \le2\pi, 0\le r \le 1, r^2+2\le z \le \sqrt{10-r^2}\}.$$ It follows that $$\iiint_E4z+2y\; dV=\int_0^{2\pi}\int_0^1\int_{r^2+2}^{\sqrt{10-r^2}}(4z+2r\sin\theta)rdzdrd\theta = \frac{19\pi}{3}.$$
• $\int 4rz+2r^2\sin\theta\; dz=4rz^2/2+2zr^2\sin\theta$ Jan 4, 2016 at 11:48