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Math Help - Integral w/ Spherical Coordinates

  1. #1
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    Integral w/ Spherical Coordinates

    I am having trouble with this triple integral. It seems so easy, but I keep getting stuck at the same spot, and I'm not sure where I am going wrong.

    Evaluate the triple integral over E, where E is bound by the sphere (x^2) + (y^2) + (z^2) = 9 in the first octant. The density function is given by:
    exp(sqroot((x^2) + (y^2) + (z^2))).

    So by using spherical coordinates I get the following as my limits:
    0 < theta < 2pi, 0 < phi < pi/4, 0 < rho < 3. Is this right?

    After changing variables, I get (exp(rho))*(rho^2)*sin(phi) for the density. Is that right? After this, I'm not sure where to go, since integrating (exp(rho))*(rho^2)*sin(phi) w/r/t rho doesn't seem like the easiest thing to do. Maybe I need to switch my order of integration?

    Any ideas? Thanks.
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  2. #2
    MHF Contributor matheagle's Avatar
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    \int_0^a \rho^2 e^{\rho} d\rho is just parts twice.

    You didn't give me the whole problem, but z>0 is 0<\phi <\pi/2.

    Likewise, the first octant gives, x>0, y>0, so 0<\theta <\pi/2.

    This should factor into three integrals.
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