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Math Help - integral set up..

  1. #1
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    integral set up..

    Find the volume of the solid bounded above by the paraboloid z = 16 - x^2 - y^2 and bounded below by x^2 + y^2 + z^2 =16.

    *note, set up the integral in rectangular, cylindrical, and spherical coordinates only. Do not evaluate the integral
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by boousaf View Post
    Find the volume of the solid bounded above by the paraboloid z = 16 - x^2 - y^2 and bounded below by x^2 + y^2 + z^2 =16.

    *note, set up the integral in rectangular, cylindrical, and spherical coordinates only. Do not evaluate the integral
    Let's do some examining prior to setting up the integration.

    At z=0, we see that both equations take on the form of x^2+y^2=16

    This implies that our solid is bounded as so:

    In Rectangular: -4\leq x\leq 4, \ \ -\sqrt{16-x^2}\leq y\leq \sqrt{16-x^2}, \ \ -\sqrt{16-x^2-y^2}\leq z\leq 16-x^2-y^2

    In Cylindrical: 0\leq r\leq 4, \ \ 0\leq \vartheta\leq 2\pi, \ \ -\sqrt{16-r^2}\leq z\leq 16-r^2

    In Spherical: 0\leq\vartheta\leq 2\pi, \ \ 0\leq\varphi\leq \pi, \ \ 0\leq \varrho\leq 4

    Hopefully, you can take it from here.

    --Chris
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