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Find the volume of the region lying inside all three of the circular cylinders $$x^2+y^2=a^2,$$ $$x^2+z^2=a^2,$$ $$y^2+z^2=a^2$$ Hint: Make a good sketch of the first octant part of the region, and use symmetry whenever possible.

I have trouble in identifying the function to integrate and the boundary of the integral. Can anyone help me?

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  • $\begingroup$ The integrand is $1$ and the domain is the intersection of the inside of the cylinders. $\endgroup$ – Yves Daoust Nov 8 '14 at 9:42
  • $\begingroup$ This comment is to link this post as one of the (abstract) duplicates to the current choice of mother/target post, which merit is not in the content nor being the oldest but merely having an existing link. $\endgroup$ – Lee David Chung Lin Jan 22 '19 at 12:05
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3 CYL INTRXN

Google for images of intersection of cylinders, above site is one useful.

First obtain intersection area/volume of two cylinders as practice , before going to the more general three mutually perpendicular tubes case. I could build a 3D model in steel. It has cut portions of sine curve at endpoints. Deciding start and end points in one octant is central task here.

EDIT1: With two cylinder intersection the plane of intersection is $ x=y $. Now choose a part of it.

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Hint - Your shadow in xy plane is x^2+y^2=a^2 chop in up into 8 parts say take a piece from theta varying 0 to pi/4 .See which cylinder u hit when drawing a ray upwards . u can do this by seeing which i down and which is up by plugging in some value in( y less than x) .

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