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Math Help - Volume of three intersecting cylinders

  1. #1
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    Volume of three intersecting cylinders

    Hi.
    I've been asked to find the volume of the region bounded by the 3 inequalities x^2+y^2<r^2, \, x^2+z^2<r^2, \, y^2+z^2<r^2. This is, three cylinders of radius r. Using a triple integral to find the body seems to be the 'natural way', but I can't see how to calculate
    \iiint \left[x^2+y^2<r^2, \, x^2+z^2<r^2, \, y^2+z^2<r^2 \right] \, dx \, dy \, dz

    Thanks!
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  2. #2
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    As a starting point, look at equation (19) on this page. The pictures between equations (17) and (18) on that page illustrate how the integral is formed.
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  3. #3
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    So the answer is 8(2-\sqrt{2})r^3?
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