Integral of a function on a cylinder

  • Thread starter brainslush
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  • #1
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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


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


The Attempt at a Solution


I'm not quite sure what to do but I give it a try.

Determine the Jacobian...

[itex]\frac{(\partial(x,y,z))}{(\partial(a,\Theta,z))}[/itex] = a

By change of variables one gets:

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

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:

Answers and Replies

  • #2
2,571
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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??
 
  • #3
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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:
  • #4
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After deeper consideration the only reasonable posibility is a surface integral.
In that case I do following:
One knows that [itex]r^{\rightarrow}(\theta,z) = (a*cos(\theta), a*sin(\theta), z)[/itex]

then one gets

[itex]\left\|\frac{\partial r}{\partial \theta} \times \frac{\partial r}{\partial z}\right\| = a[/itex]

finally

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

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