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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Volume between cylinders

**Physics Forums | Science Articles, Homework Help, Discussion**