How can the heat equation be derived for a long circular cylinder?

[BustedBreaks]

Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here r=\sqrt{x^{2}+y^{2}} is the cylindrical coordinate. From the three dimensional heat equation derive the equation u_{t}=k(u_{rr}+\frac{u_{r}}{r}).

My book describes how the general heat equation is derived, but I'm having trouble incorporating the equation for r and in general understanding these concepts.

The book starts out with:

H(t)=\int\int\int c\rho u dxdydz which I'm somewhat confident I can change to

H(t)=\int\int\int c\rho u dxdydr because of the equation with r

Then the book has
\frac{dH}{dt}=\int\int\int c\rho u_{t} dxdydr which makes sense to me

then after this I am a bit confused. I don't really know what to do next in terms of answering the question.

 

[ideasrule]

This can be done very easily, by rewriting the heat equation in spherical coordinates and taking advantage of the symmetry of the situation. A little bit of re-arranging gives you the equation.