Hi, I'm having trouble with the following question.(adsbygoogle = window.adsbygoogle || []).push({});

Q. Use triple integrals andcartesiancoordinates to find the volume common to the intersecting cylinders x^2 + y^2 = a^2 and x^2 + z^2 = a^2.

This question pops up in basically every introductory calculus text. I've seen it before but I simply don't know how to set up the integral to do this question.

I know that the integral is of the form:

[tex]

V = \int\limits_a^b {\int\limits_{h_1 \left( x \right)}^{h_2 \left( x \right)} {\int\limits_{g_1 \left( {x,y} \right)}^{g_2 \left( {x,y} \right)} {dzdydx} } }

[/tex]

The order of integration is nominal, I can change the projections if needed but I just chose z then y then x so that I have something to begin with.

In mos of the questions that I've done, the range of z has simply been an fairly simple and easy to see (eg. 0 <= z <= 8) but that isn't the case in this question. Since I can't find the range of z values I'll start with find the projection of the region onto the x-y plane.

x^2 + y^2 = a^2 and x^2 + z^2 = a^2. When the cylinders intersect I'll obtain x^2 + y^2 = x^2 + z^2 => y^2 = z^2? I'm having trouble visualising the region and finding the terminals on the volume integral. Can someone please help me out?

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# Volume between cylinders

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