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Math Help - Volume of sphere minus cylinder

  1. #1
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    Volume of sphere minus cylinder

    The sphere has a radius of 5

    The cylinder has a radius of 4

    My integral for my circle is \int_0^5(25-x^2)dx

    My integral to subtract from that is \int_0^4(25-x^2)dx

    I do not think this is right

    any help?
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  2. #2
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    Quote Originally Posted by silencecloak View Post
    The sphere has a radius of 5

    The cylinder has a radius of 4

    My integral for my circle is \int_0^5(25-x^2)dx

    My integral to subtract from that is \int_0^4(25-x^2)dx

    I do not think this is right

    any help?
    First think of this in the upper xy-plane.
    The equation of the top sphere is z=\sqrt{25-x^2-y^2}.
    If you compute, \iint_R \sqrt{25-x^2-y^2} ~ dA where R = \{ (x,y) | x^2+y^2 \leq 16 \} you get the area of the cylinder below sphere.
    Thus, the total volume this cycliner-like piece takes up is double that integral.
    Once you know this number you can finish the problem.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    First think of this in the upper xy-plane.
    The equation of the top sphere is z=\sqrt{25-x^2-y^2}.
    If you compute, \iint_R \sqrt{25-x^2-y^2} ~ dA where R = \{ (x,y) | x^2+y^2 \leq 16 \} you get the area of the cylinder below sphere.
    Thus, the total volume this cycliner-like piece takes up is double that integral.
    Once you know this number you can finish the problem.
    I can kind of see how that works but I would never come up with something like that myself(And honestly I don't know where to start to compute that). The way we were taught in class is how I have tried to set it up.

     2\pi\int_0^5(25-x^2)dx - 2\pi\int_0^4(25-x^2)dx  = 29.3215314335

    or

     2\pi\int_4^5(25-x^2)dx = 29.3215314335
    Would this be correct?

    Thank you for your help
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  4. #4
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    Consider y=\sqrt{25-x^2} the top half of the circle.
    If you compute, 2\pi \int_{-3}^3 xy dx you get the volume of the cyclinderical piece.
    Attached Thumbnails Attached Thumbnails Volume of sphere minus cylinder-picture.jpg  
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  5. #5
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    I got it today in class, thank you for your help...

    Problem was i was trying to rotate the shape around the wrong axis
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