# Integral of a function on a cylinder

## 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.

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

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??

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

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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

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