Calculate the flux of the vector field by using divergence theorem problem

Question

Calculate the net flux of the vector field $F(x, y , z) = (x, y , 3)$ out of the region given by the inequalities$\sqrt{x^2 + y^2} ≤ z ≤ \sqrt{2 − x^2 − y^2}$

But the correct answer is $\frac{8π}3(\sqrt{2} − 1)$. What am I doing wrong?

Answer 1

The two cones $z=\sqrt{x^2 + y^2}$ and $z=\sqrt{2 − x^2 − y^2}$ intersect along the plane $z=1$ and their intersection is a circle of radius $1$. It follows that the limits of integration for $r$ should be from $0$ to $1$ (not $\sqrt{2}$). Therefore, in your work, the final integral should be $$2\pi\int_0^1(2\sqrt{2-r^2}-2r)rdr=4\pi\left[-\frac{(2-r^2)^{3/2}}{3}-\frac{r^3}{3}\right]_0^1=\frac{8π}3(\sqrt{2} − 1).$$