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Math Help - surface integrals for sphere

  1. #1
    n22
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    surface integrals for sphere

    Hello,

    I am having trouble integrating this one ...especially because there is (sinθsinφsin2φ)/2.. when one tries to integrate the following ..
    ∫∫2cosφsinθsinθdφdθ
    inner integral:
    φ:-pi/2 to to pi/2

    theta :0 to pi/2
    How does one integrate this... ??

    thanks
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  2. #2
    MHF Contributor

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    Re: surface integrals for sphere

    That " sin(\theta)sin(\theta)" looks peculiar. Is it really [tex]sin^2(\theta)[itex] or did you mean to write "[tex]sin(\theta)cos(\theta)[/itex]"?

    Either way, it's a pretty straightforward integral.
    \int_0^{\pi/2}\int_{-\pi/2}^{\pi/2} 2cos(\phi)sin^2(\theta)d\theta d\theta d\phi= 2\int_0^{\pi/2}cos(\phi)d\phi \int_{-\pi/2}^{\pi/2} sin^2(\theta) d\theta
    Of course, the integral of "[tex]cos(\phi)[/itex]" is " sin(\phi)" and to integrate " sin^2(\theta)" use the identity sin^2(\theta)= \frac{1}{2}(1- cos(2\theta)).

    If you intended sin(\theta)cos(\theta) rather than sin^2(\theta),
    \int_0^{\pi/2}\int_{-\pi/2}^{\pi/2} 2cos(\phi)sin(\theta)cos(\theta)d\theta d\theta d\phi= 2\int_0^{\pi/2}cos(\phi)d\phi \int_{-\pi/2}^{\pi/2} sin(\theta)cos(\theta) d\theta
    Of course, the integral of " cos(\phi)" is " sin(\phi)" and to integrate " sin(\theta)cos(\theta)" use the substitution u= sin(\theta).
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