Vector field surface integral over cylinder gives zero but should be 4π

[cpfoxhunt]

Homework Statement

This is annoying me because I am so clearly being a muppet somewhere. I need to integrate the vector field (x,-y,z).(vector)ds over the surface of a cyliner x^2 + y^2 < 4 (or equal to) and z is between 0 and 1.

The Attempt at a Solution

I have to do it both with and without divergence. With divergence, div(a) = 1, integral over a nice volume, gives me 4Pi, as O a, (integrating 2*r drd(theta)dz, with the specified limits.)

Trying to do it as a surface integral is driving me nuts. I end up integrating r^2(sin^2(x) - cos^(x) between the limits x = 0 and x = pi. as this integrates to sin(x)cos(x), this makes the entire integral zero.

So where am I going wrong?

 

[Dick]

I don't think you included the contributions from the flat ends of the cylinder, did you?