A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ where $c - d > a + b$ and $d > 0$.
Calculate the total area of the propeller as shown by the shaded region and calculate the average temperature of the propeller, given by $T (r, α) = C/r$, where $C$ is a constant.
Have absolutely no idea how to solve this.
On
Update; my previous answer was wrong.
The area element in polar coordinates is given by $${\rm d(area)}=r\>{\rm d}(r,\phi)\ .$$ Therefore the area $A$ of your propeller is given by $$A=\int_0^{2\pi}\int_{a+b\cos(4\phi)}^{c+d\cos(3\phi)} r\>dr\ d\phi\>,$$ and the "heat content" $H$ computes to $$H=\int_0^{2\pi}\int_{a+b\cos(4\phi)}^{c+d\cos(3\phi)} {C\over r} r\>dr\ d\phi\ .$$ Finally the average temperture is obtained as $$T_{\rm mean}={H\over A}\ .$$ The following figure shows the mathematica output for this problem. Your $C$ has been replaced by $p$ therein.
(This answer was dead wrong. Unfortunately it cannot be deleted. See my new answer below.)