# Evaluate the surface integral over the surface of the sphere

Evaluate the surface integral: I = $\int \int_S \ F \cdot n \ dS$ over the surface of the sphere described by $x^2 + y^2 + z^2 = 16$ if $F = xy^2i + yz^2j+ zx^2k$

Attempt at problem: I use the divergence theorem: $\int \int_S \ F \cdot n \ dS = \int \int \int_V \nabla \cdot F \ dV$ I found $\nabla \cdot F$ to be $x^2 + y^2 + z^2 + 2xz + 2xy + 2yz$. All of this implies $\int \int \int_V x^2 + y^2 + z^2 + 2xz + 2xy + 2yz \ dV$ Since there is a sphere, the integral becomes (converting rectangular to spherical coordinates) $\int_0^{\pi} \int_0^{2\pi} \int_0^4 [r^2(r^2 + 2r^2(\cos{\theta}\sin{\phi}\cos{\phi} + \cos{\theta}\sin{\theta}\sin^2{\phi} + \sin{\theta}\sin{\phi}\cos{\phi})]$

• You have miscalculated the divergence of $F$. – Paul Sinclair Oct 19 '15 at 1:13
• How so? I believe I also should have got a numerical value. – WAS Oct 19 '15 at 1:22
• $\nabla \cdot F = x^2 + y^2 + z^2$ I have no idea where you got the other terms from. – Paul Sinclair Oct 19 '15 at 1:26