# Thread: Volume of three intersecting cylinders

1. ## Volume of three intersecting cylinders

Hi.
I've been asked to find the volume of the region bounded by the 3 inequalities $\displaystyle x^2+y^2<r^2, \, x^2+z^2<r^2, \, y^2+z^2<r^2$. 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
$\displaystyle \iiint \left[x^2+y^2<r^2, \, x^2+z^2<r^2, \, y^2+z^2<r^2 \right] \, dx \, dy \, dz$

Thanks!

2. 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.

3. So the answer is $\displaystyle 8(2-\sqrt{2})r^3$?

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# volume of 3 intersecting cylinders

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