Using Divergence theorem to calculate integral

Question

Question: Use Divergence Theorem to compute $\int \int_D^\ (F.n) dS$ where S is bounded by $z=4-x^2-y^2$, z=0 and z=1, and the force field is $F = (z^3,x^2y,y^2z)$

Attempt to solve: $$div(F) =x^2+y^2 $$ Using cylindrical coordinates $\int \int_D^\ (F.n) dS=\int\int\int_D div(F) dV$ $=\int_0^{2\pi}\int_0^2\int_0^{4-r^2}(z^2r)\,dz\,dr\,d\theta=\frac{32}{3}\pi $ However, the correct answer is 1. What am I doing wrong? Any help is much appreciated.

Answer 1

The correct setting for the integral should be

$$=\int_0^{2\pi}d\theta\int_0^1 dz \int_0^{\sqrt{4-z}}r^3\,dr$$

indeed

• $z$ varies from 0 to 1
• for $z$ fixed, $r$ varies from 0 to $\sqrt{4-z}$
• the divegence is $r^2$