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Math Help - triple integral

  1. #1
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    triple integral

    Hey.

    What's the answer for this:

    \int_{0}^{r}dr  \int_{0}^{2\pi}rd\phi  \int_{0}^{l}dl \rho (1-r/R)
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  2. #2
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    \rho l2\pi r^{2}[\frac{1}{2}-\frac{r}{3R}]
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  3. #3
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    Quote Originally Posted by jacek View Post
    Hey.

    What's the answer for this:

    \int_{0}^{r}dr  \int_{0}^{2\pi}rd\phi  \int_{0}^{l}dl \rho (1-r/R)
    Actually, that is a very badly posed integral. It is never a good idea to use the same letter to represent the variable of integration and a limit of integration and that is done twice here. Other than that, it's pretty close to trivial. We can "separate" those integrals and do them separately:
    \left(\int_0^r (1- r/R)r dr\right)\left(\int_0^{2\pi} d\phi\right)\left(\int_0^l dl\right).

    And, of course, \int_0^{2\pi} d\phi and \int_0^l dl are trivial!

    \int_0^r (1- r/R)r dr= \int_0^r rdr- \frac{1}{R}\int_0^r r dr should also be easy though not "trivial".
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