Flux through a section of a sphere

[Feodalherren]

Homework Statement

Find the flux of F=<y,-x,z> through the piece of ρ=2 that lies above z=1 and is oriented up.

Homework Equations

The Attempt at a Solution

$S = < x, y, \sqrt{4-x^{2}-y^{2}} >$

Take Find Sx and Sy, cross them and end up with:

$dS = < \frac{x}{\sqrt{4-x^{2}-y^{2}}}, \frac{y}{\sqrt{4-x^{2}-y^{2}}}, 1 >$

Z is positive, orientation is OK.

F dot dS = $\sqrt{4-x^{2}-y^{2}}$

Therefore the integral should be

$\int^{2\pi}_{0}\int^{\sqrt{3}}_{0} r\sqrt{4-r^{2}}drd\theta$

= $\frac{4\pi}{3} (\sqrt{32} -1)$

Incorrect.
The correct answer is

$\frac{14\pi}{3}$

 

[HallsofIvy]

All I can say is that you have clearly integrated wrong. Since you don't show how you did that integral, I cannot say more.

 

[Feodalherren]

$\int^{2\pi}_{0} \int^{\sqrt{3}}_{0}r \sqrt{4-r^{2}}dr d\theta$

= $2\pi \int^{\sqrt{3}}_{0}r\sqrt{4-r^{2}}dr$

=$\pi \int^{4}_{1}u^{1/2}du$

= $\frac{2\pi}{3}(4^{3/2}-1)$

I see what I did. Thanks.