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Homework Help: Integral of a function on a cylinder

  1. Oct 5, 2011 #1
    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
    [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]


    3. 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: Oct 5, 2011
  2. jcsd
  3. Oct 5, 2011 #2
    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??
     
  4. Oct 5, 2011 #3
    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
  5. Oct 5, 2011 #4
    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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