What Went Wrong in My Verification of the Divergence Theorem?

[khemist]

Homework Statement

Let the surface, G, be the paraboloid z = x^2 + y^2 be capped by the disk x^2 + y^2 \leq 1 in the plane z = 1. Verify the Divergence Theorem for \textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k}

Homework Equations

I have solved the problem using the divergence theorem, that is no problem. However, I am having trouble verifying, where I used the formula \iint_G \textbf{F} \bullet d\textbf{S}

The Attempt at a Solution

My projection on the xy-plane is a circle with the equation x^2 + y^2 = 1. My n1 (normal vector), pointing from the plane z=1, is simply k. The n2, coming out of the surface z = x^2 + y^2, I got is 2x i + 2y j - k.

I converted my limits to polar coordinates, to get an integral

\int_0^{2\pi}\int_0^1 (4r^3(1+cos2\theta) - r^5(1-cos2\theta)) dr d\theta

However, when I solve this, I get \frac{2\pi}{3} when it should be \frac{5\pi}{2}.

Any ideas on what I have done wrong? thanks for the help

 

[vela]

It's hard to say when you don't show intermediate steps. Your normals look fine.

I'll note that when I calculated the surface integral and the volume integral, I found them both equal to 4\pi/3, not 5\pi/2.