Divergence theorem

Question

I have to use the divergence theorem to solve

$\iint F \cdot ds$ where $F(x,y,z)=x^3 \hat{\imath}+y^3\hat{\jmath}+z^3\hat{k}$ and $S$ is the surface of the solid rounded by the cylinder $x^2+y^2=1$ and the planes $z=0$ and $z=2$

any help ? Thanks

Answer 1

Since you've computed the divergence, the next step is to take the integral of the divergence over the region. In this case, that is $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = \int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_0^2 \nabla \cdot F(x,y,z)\,dz\,dy\,dx $$ But perhaps you'll find it easier to change this integral over to cylindrical coordinates instead.