Circle surface area averaged value

Question

I have a simple function that relates a variable $y=f(x)$ to its position $x\in [0,R]$ on the radius $R$ of a circle. This function is of the form $f(x)=ax$.

How can I find the average value of $y$ over the total surface area of the circle?

Thank you

Answer 1

My understanding is $$f(x,y)=a\sqrt{x^2+y^2}$$ where $x^2+y^2\le R$.

\begin{align*} \langle f \rangle &= \frac{1}{\pi R^2} \iint_{x^2+y^2\le R} f(x,y)\, dx\, dy \\ &= \frac{1}{\pi R^2} \int_{0}^{2\pi} \int_{0}^{R} ar \, rdr\, d\theta \\ &= \frac{1}{\pi R^2} \int_{0}^{2\pi} \frac{aR^3}{3} d\theta \\ &= \frac{2aR}{3} \end{align*}