Hi Sun,

You're on the right path. To calculate this flux integral, you should first realize that parametrizing the surface Q would be hard, so it's better to try to use the Divergence Theorem and instead compute the integral of the divergence of F not over Q but over the *volume enclosed *by Q. You're right that div F = 2z. So you just need to set up the volume integral.

Note that x^{2}+y^{2}=1 determines a cylinder of radius 1. So the volume is best described in cylindrical coordinates, which is just polar coordinates in the xy-plane together with the z coordinate. If you sketch the surface Q you should be able to convince yourself that the volume V will be parametrized by theta in [0,2pi), z in [0,1], and r in [z,1]. (To see that last part, notice that z=sqrt(x^{2}+y^{2}). But x^{2}+y^{2}=r^{2} in polar coordinates, so this says z=r.)

These are your limits of integration, so you just carry out the integral of 2z in cylindrical coordinates with the cylindrical volume element r dr dtheta dz. If you do this, you do indeed get an answer of pi/2. Let me know if you need more help.

Best wishes,

Matt

B.S. Mathematics, MIT

## Comments

Thank you so much! Wow, you attended MIT? Did you know that MIT is my dream school? I'll apply there this fall after taking the SAT. But I don't have any extracurricular activities, can I still get accepted to MIT? I have some big reasons to go there, though.