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Math Help - Triple Integral Calculation

  1. #1
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    Triple Integral Calculation

    Hello,

    How do I calculate the triple integral for the solid created by the triple integral of "x dV" with limit "D", where D is bounded by paraboloid z=x^2+y^2 and the plane z=1?

    Any help is much appreciated. Thanks.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by HFLER View Post
    Hello,

    How do I calculate the triple integral for the solid created by the triple integral of "x dV" with limit "D", where D is bounded by paraboloid z=x^2+y^2 and the plane z=1?

    Any help is much appreciated. Thanks.
    The easiest way to approach this is to use cylindrical coordinates.

    Consider the cross section of the solid when z=1: We have the circle x^2+y^2=1\implies r^2=1\implies r=1, after our conversion to cylindrical coordinates.

    Thus, we can make the claim that 0\leqslant r\leqslant 1, 0\leqslant\theta\leqslant2\pi, and 1\leqslant z\leqslant r^2.

    Taking into consideration that x=r\cos\theta in the cylindrical coordinate system, we see that \iiint\limits_D x\,dV=\int_0^{2\pi}\int_0^1\int_0^{r^2}\left(r\cos  \theta\right) r\,dz\,dr\,d\theta=\int_0^{2\pi}\int_0^1\int_0^{r^  2}r^2\cos\theta\,dz\,dr\,d\theta

    Can you continue on with the calculation?
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