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Triple Integral

  1. Jul 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Find [tex]\int\int\int y^2 z^2[/tex]where E is the region bounded by the paraboloid x = 1 - y22 - z2 and the plane x = 0.

    3. The attempt at a solution

    The region is a paraboloid with vertex at x = 1, y = 0, z = 0. I chose z bounds to be between 0 and 1 - y22 - z2 for first integral. Then, I realized that since region was bounded by x = 0 plane, the y and z values would range (in polar coordinates), from 0 to 2[tex]\pi[/tex] for y (or z) and 0 to 1 for z.

    Then, upon finishing first integral for z bounds, I got (1 - y2 - z2) * y2*z2, and when converting to polar coordinates, I got,

    (1 - r2)*(r4*cos2([tex]\theta[/tex])*sin2([tex]\theta[/tex])

    I don't know how to simplify this expression so that I can integrate for theta. How do I do it?
     
  2. jcsd
  3. Jul 16, 2009 #2

    Mark44

    Staff: Mentor

    Is there a typo in x = 1 - y22 - z2 or did you mean x = 1 - 2y2 - z2?
    The usual practice is to put numerical coefficients before variables.
     
  4. Jul 16, 2009 #3
    sorry, it is meant to read x = 1 - y2 - z2
     
  5. Jul 17, 2009 #4

    Mark44

    Staff: Mentor

    You seem to be ignoring the differentials dx, dy, and dz in your first integral, and the dr [itex]d\theta[/itex] and dz in your integral converted to polar form.

    Due to the symmetry of your region and the integrand, you can take [itex]\theta[/itex] between 0 and [itex]\pi/2[/itex], and multiply the resulting integral by 4.

    It would be helpful to see your integral with limits and with differentials. You can see my LaTeX code just by clicking it.
    [tex]
    \int_{z = ?}^{?} \int_{\theta = ?}^{?} \int_{r = ?}^{?} <integrand> r dr d\theta dz
    [/tex]

    You'll need to fill in the lower and upper limits of integration, and the integrand will need to be converted to polar form as well.
     
  6. Jul 17, 2009 #5
    [tex]

    \int_{r = 0}^{1} \int_{\theta = 0}^{2 \pi} (1 - r^2) r^4 (\cos(\theta))^2 (\sin(\theta))^2 d\theta dr

    [/tex]
     
  7. Jul 17, 2009 #6

    Mark44

    Staff: Mentor

    You have skipped a step. Let's start from the triple integral that I provided.
     
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