# Representing a region as limits of a volume integral

1. Aug 28, 2011

### msslowlearner

1. The problem statement, all variables and given/known data

i have the region given as being bounded by x2+y2=4 and z=0 and z=3.
this problem asks to prove gauss divergence theorem for a given vector F

2. Relevant equations
3. The attempt at a solution
As for the volume integral, i had no problem. But for the surface integral, how many surfaces are there actually? How are the normals represented?

I assumed it is a cylinder(Is it??).. and i got the normal vectors to the top surface and the bottom surfaces to be k and -k respectively. But these cancel out effectively. The problem here is i dont know how to represent any other surface, if any.. i really do not know to find the others. could you give me an idea on how to go about this ? i mean, finding the surfaces and limits..

2. Aug 28, 2011

### LCKurtz

Yes, the side surface is a cylinder. You might parameterize the surface x2+y2 = 4 for z between 0 and 3 like this:

R(θ,z) = <2cos(θ), 2sin(θ),z>, 0 ≤θ≤ 2pi, 0≤z≤3

3. Aug 28, 2011

### HallsofIvy

Staff Emeritus
There are three "sides", the cylindrical $x^2+ y^2= 4$ and two circular ends at z= 0 and z= 3.
For those good parametric equations would be $x= r cos(\theta)$, $y= r sin(\thet)$, with r from 0 to 2 and $\theta$ from 0 to $2\pi$, z= 0 or 3.