How Does Radial Temperature Vary in a Thermally Conductive Hollow Cylinder?

[Domisterwoozy]

Homework Statement

A hollow cylinder of length L has inner radius, r1, outer radius, r2 , and
thermal conductivity, k. It is in dynamic thermal equilibrium, with its interior
held at temperature T1 and its exterior at a different temperature, T2. What is the
radial temperature dependence, T(r), within the cylinder, r1 ≤ r ≤ r2 ?

Homework Equations

Heat Current = KA(T-T)/L

The Attempt at a Solution

I know heat current out equals heat current in and than you solve for T. However, I cannot figure out how to calculate the heat currents since the area is changing.

 

[ideasrule]

You need to use calculus. Consider a very thin shell surrounding the center of the cylinder, of thickness dr. The area of one side isn't very different from the area of the other, so you can use that heat conduction equation.