1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral of a function on a cylinder
Loading...