Volume of three intersecting cylinders

**DavidEriksson** · February 6th 2010, 11:14 AM

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!

**Opalg** · February 6th 2010, 01:06 PM

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.

**DavidEriksson** · February 7th 2010, 12:27 AM

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

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