I need help with a math problem for math club tomorrow. Its a fun presentation, but I need help with this one problem. Thanks in advance!
Given the transient point source solution valid within an isotropic half space
$T=\frac{q}{2\pi \:kr}erfc\left(\frac{r}{2\sqrt{\alpha \:t}}\right),\:dA\:=\:r\:dr\:d\theta$
derive the expression for the transient temperature rise at the centroid of a circular area ($\pi a^2$) which is subjected to a uniform and constant heat flux q. Superposition of point source solutions allows one to write
$T_0=\int _0^a\:\int _0^{2\pi }\:T\:dA$
Here's an exact photo, just in case I wasn't clear originally :).
https://i.stack.imgur.com/aIaZJ.png
To be clear, I think that one would use the derivative in terms of A and substitute, but I'm not completely sure where to go from there, so it would be very helpful if someone was able to help me be able to solve it. I know what the error function is, but I'm not sure how to use it in this problem. To be clear, I don't need help with my homework, its just for an extracurricular presentation, and I want to be able to teach my fellow math club friends about how to solve an equation like this. It would further our education in the subject, and I think everyone would benefit as a result. Thanks again! :)