Heat flow through a rod with varying cross-section

[Wiz]

Hi,
I have to find the rate of heat flow through a cross-section of the rod in steady state as shown in the figure.Thermal conductivity of the rod is K.
I have used a $(dollar) sign for theta.
The formula i know is ΔQ/Δt = [KA($2 - $1)] /x
where ΔQ/Δt is the heat current,where A is area of cross section and x is the thickness.
since the area of cross section varies.i think integration will be used...
can ANYBODY GUIDE ME THROUGH..??
[NOTE - THE FIGURE READ THAT $2 > $2 , which is a mistake and the correction is $2 > $1.]
Thanks,

 

[davidY]

1. assume that the steady state
2. get a thin slice of tube
3. since steady state: heat enter=heat exit
4. this means H is constant over all x
5. our equation:

H=-kA(dT/dx)​

H=-k*pi*r^2*(dT/dx)​

7. well, pi is constant over all x
8. well, r is not constant over all x
9. good luck integrating...

 

[Wiz]

??

how do i relate x and r...
there are 3 variables and we need only 2 for integration.
hope u understand,
wiz

(btw - all the steps u wrote were know to me already0