Find the average temperature of the body?

Question

A body occupies the region in space inside the cylinder x^2+y^2=4, above the xy plane, and below the surface z=2+x. The temperature at the point (x, y, z) in the body is given by T(x, y, z)=2x. Find the average temperature of the body.

Answer 1

Oh goodness gracious.

So know that this is possible and doable and even kind of pretty.  It just is going to take a lot of steps as far as I can see.  I'm not going to do a step by step explanation right now because I don't think I have time tonight, but I'll give some pointers.

First, start by setting up the problem so that you integrate the z variable first, then y and end with x.  To find the temperature you are going to be integrating the function 2x over the volume of the shape described which has some lovely equations to help you define it.  To find the volume of the darned thing you're going to need to integrate the function f(x) = 1 over that same volume.

Along the way you will come across a few integrations that don't look like they will be nice.  They are fine, there are a few ways you could solve it, but the easiest way I see to solve it is to immediately set x = 2sin(θ).  It comes in handy.  Just remember that means that dx = 2cos(θ)dθ.

Let me know if that helps and I'll try to give a step by step tomorrow.