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Math Help - Cylindrical coordinates..

  1. #1
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    Cylindrical coordinates..

    Use cylindrical coordinates to find the volume of the solid in the first octant inside of the sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 2x

    Note: set up the integral only, do not evaluate.
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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
    Use cylindrical coordinates to find the volume of the solid in the first octant inside of the sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 2x

    Note: set up the integral only, do not evaluate.
    The region they have in common is the cross sectional area of the cylinder (x-1)^2+y^2=1, thus, we see that 0\leq x\leq 2, -\sqrt{1-(x-1)^2}\leq y\leq \sqrt{1-(x-1)^2}, and the height of the solid would be defined by -\sqrt{4-x^2-y^2}\leq z\leq \sqrt{4-x^2-y^2}

    In cylindrical coordinates, we can see that 1\leq r\leq 2, 0\leq\vartheta\leq2\pi, and -\sqrt{4-r^2}\leq z\leq \sqrt{4-r^2}

    After a little more work, we see that the volume could be represented as 2\int_0^{2\pi}\int_1^2 \int_0^{\sqrt{4-r^2}}r\,dz\,dr\,d\vartheta

    Does this make sense?

    [I'm pretty sure my limits on r are correct...]

    --Chris
    Last edited by Chris L T521; August 2nd 2008 at 08:32 PM. Reason: typo??
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  3. #3
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    Thanks very much Chris! Perfect explanation!
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