Take the 2-minute tour ×  I am having trouble simplifying the
following calculation:  Let $S=\{(x,y,z)|x^2+y^2+z^2=25,-4\leq
x,y,z \leq 4\}$ and $F=(x^3,y^3,z^3)$. I am asked to evaluate
the surface integral $\int_{{S}}^{{}} F\cdot n$ where $n$ is the
outward pointing unit normal.  Attempt I 'managed' to calculate
the integral in the following way: first I completed S to a
ball, used the divergence theorem, then subtracted the
leftovers. However this led to a lot of what seems to be
needless complexity in the calculation, and the integrals turned
out to be very complex. I think my end result was wrong, too; I
got $$15000 \pi -6 \times 3/5 \times 512 \pi -6 \times 9 \times
4^3 \times \pi$$  So I guess I'm looking for an easier, more
straightforward way to do this which wouldn't involve as many
complex calculations.   share|improve this question add comment
2 Answers  up vote 0 down vote accepted  It is obvious the flux
of the whole sphere is $\iiint_V 3r^2 dV = \frac{12}5\pi R^5 =
7500\pi$. I guess there's no