Integral of a function on a cylinder

[brainslush]

Homework Statement

Find the integral of the function x^2 on a cylinder (excluding button and top)
x^2 + y^2 = a^2,
0 <= z <= 1

Homework Equations

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

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.

 

[flyingpig]

Find the integral of the function x^2 on a cylinder (excluding button and top)

Technically this question 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??

 

[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

 

[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}