Question : If $T(r) = \frac{T(0)}{r^3}$ is the temperature profile in the region R, then use the previous results to calculate the average temperature in R when $T(0) = 1000$. Verify that the average temperature is between the minimum and maximum temperatures in R.
R is the region bounded by the curve $C = C_1 \cup C_2$.
$C_1$ is a semicircle with centre at the origin $O$ and radius $9/5$. $C_2$ is part of an ellipse with centre at $(4, 0)$, horizontal semi-axis $a = 5$ and vertical semi-axis $b = 3$.
Previous Results
$\int_C\mathbf v \cdot d \mathbf r$ (the line integal of $\mathbf v$ along $C$ where $\mathbf{v} = \frac{1}{2} (-y\mathbf{i} + x\mathbf{j})) = \frac{-81\pi}{5} + 15\cos^{-1}(\frac{-4}{5}) + \frac{36}{5}$
Cartesian Equation for ellipse: $\frac{(x-4)^2}{25}$ + $\frac{y^2}{9}$ = 1
Polar Equation: $9 + 4r\cos(\theta) = 5r$
$\iint_R\frac{1}{r^3} dA = \frac{8}{9}$, Where $r = \frac{9}{5-4\cos(\theta)}$
Any help with this would be much appreciated, Thanks!