# Integral of a function on a cylinder

1. Oct 5, 2011

### brainslush

1. The problem statement, all variables and given/known data
Find the integral of the function x^2 on a cylinder (excluding button and top)
x^2 + y^2 = a^2,
0 <= z <= 1

2. Relevant equations
$\int\int\int x^{2} dx dy dz$
$x = a * cos \Theta$
$y = a * sin \Theta$
$z = z$

3. The attempt at a solution
I'm not quite sure what to do but I give it a try.

Determine the Jacobian...

$\frac{(\partial(x,y,z))}{(\partial(a,\Theta,z))}$ = a

By change of variables one gets:

$\int^{1}_{0}\int^{2\Pi}_{0}\int^{a}_{0} a^{3}*cos^{2}(\Theta) da d\Theta dz = \frac{\Pi}{4}*a^4$

Is this right or am I wrong? I guess I got stuck in the middle between parametrization, divergence, stokes, greens thm and simple integration.

Last edited: Oct 5, 2011
2. Oct 5, 2011

### flyingpig

Technically this quesiton feels incomplete.

So you are looking for volume with f(x,y,z) = x^2 ?

Why do you only have on integral with three differentials??

3. Oct 5, 2011

### brainslush

Sorry that's all we got. That's the reason why I'm not sure what to do.

Oh my fault. Yes it should be three integrals.

It looks more like a surface integral to me but this means my calculations up there are incorect

Last edited: Oct 5, 2011
4. Oct 5, 2011

### brainslush

After deeper consideration the only reasonable posibility is a surface integral.
In that case I do following:
One knows that $r^{\rightarrow}(\theta,z) = (a*cos(\theta), a*sin(\theta), z)$

then one gets

$\left\|\frac{\partial r}{\partial \theta} \times \frac{\partial r}{\partial z}\right\| = a$

finally

$\int_{S}x^{2}dS =\int^{2\Pi}_{0}\int^{1}_{0} a^{3}*cos^{2}(\theta) dz d\theta = \pi a^{3}$
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