Flux through cylindrical wedge

[TopCat]

Homework Statement

Given $$\textbf{F} = x\textbf{i} + y\textbf{j} + z\textbf{k}$$, what is the flux of $$\textbf{F}$$ through the cylinder $$x^2 + y^2 =1$$ bounded by the planes $$z=0, x+y+z=2$$.

The Attempt at a Solution

By Gauss' Theorem, $$\int\int_{S}\textbf{F}\cdot d\textbf{S} = \int\int\int_{V}(\nabla\cdot \textbf{F})dV$$

But $$\nabla\cdot \textbf{F}=3$$, so the flux through the surface equals 3 times the enclosed volume. Using cylindric coordinates to calculate the volume from the integrals or using the fact the the volume is half that of a cylinder of radius 4, the volume is $$2\pi$$ and that gives $$6\pi$$ as the flux.

However, on a multiple choice test I just took, the answers offered were 0, $$\pi$$, $$2\pi$$, $$4\pi$$, and $$10\pi$$. Where did I make a mistake?

 

[gomunkul51]

Hello!

I did the volume integral with the boundaries:

$$\int\int\int_{V}(\nabla\cdot \textbf{F})dV$$

z = [0, 2 - x - y]
r = [0, 1]
theta = [-pi, pi]

x = r cos(theta)
y = r sin(theta)

and I also got the answer of 6pi :)