Representing a region as limits of a volume integral

[msslowlearner]

Homework Statement

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

Homework Equations

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 don't 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..

 

[LCKurtz]

Homework Statement

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

Homework Equations

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 don't 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..

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

 

[HallsofIvy]

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.