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Math Help - Triple Integral Question

  1. #1
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    Triple Integral Question

    I have the following problems:

    (a) Evaluate \int_{R^{3}}  e^{{-(x^2+y^2+z^2)}}dV

    (b) Evaluate \int_{R^{3}}   e^{{-(x^2+2y^2+3z^2)}}dV

    I know how I am basically going to approach these (I am going to use spherical coordinates).

    However, in other problems I was either given or could figure out the limits of \theta, \phi, and \rho. Here, I am at a loss. Any thoughts about what I should use for the limits of each of them?
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  2. #2
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by Redding1234 View Post
    I have the following problems:

    (a) Evaluate \int_{R^{3}}  e^{{-(x^2+y^2+z^2)}}dV

    (b) Evaluate \int_{R^{3}}   e^{{-(x^2+2y^2+3z^2)}}dV
    Can't you just do...

    \int_{R^{3}}  e^{{-(x^2+y^2+z^2)}}dV = \int\int\int e^{-x^2}e^{-y^2}e^{-z^2}dzdydx = \int \int [e^{-x^2}e^{-y^2}]\int e^{-z^2}dzdydx

    Since [e^{-x^2}e^{y^2}] can be treated as a constant when you are integrating w.r.t. z...

    Continuing in the manner will give you your solution.
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  3. #3
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    Thanks for that approach! Do you know how the bounds of each variable should be handled? That's something I'm still not sure about.
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  4. #4
    Super Member Anonymous1's Avatar
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    I don't think it is a definite-triple integral. Just find the integral function, i.e., indefinite.

    BTW have you ever heard of the error function? Because the solution to your first integration is [e^{-x^2}e^{-y^2}]\frac{1}{2}\sqrt{\pi} erf(z)
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