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There is a quarter circle $O$ of radius $1$ centered on $\left (0,0 \right)$. There is a point $M$ at $\left ( \frac{\sqrt{2}}{2},0 \right ).$

I want to calculate the average of the distances between all points in the quarter circle and the minimum to the center or $M$.

In other words, $$ \text{avg}(\min(\text{dist}(p, \text{center}), \text{dist}(p, M)))$$ for all points $p$ in the area of quarter circle $O$.

So far I know that the quarter circle will be divided along $x = \frac{\sqrt{2}}{4}$, where everything to the left of the line is closer to the center, and everything to the right is closer to $M$.

I also know that I will probably have to divide the right side with a curve with center $0$ and radius $1/2$ to make calculus easier. Then, I will need to do polar calculus to actually calculate the average distance, which is the part that I don't know how to do.

Any help is appreciated! Thanks!

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  • $\begingroup$ How is the quarter circle aligned? $\endgroup$
    – Parcly Taxel
    Commented May 1, 2021 at 1:21
  • $\begingroup$ Right, I should have clarified. The quarter circle goes from -pi/4 to pi/4. So it's split in half by y=0. $\endgroup$
    – porpoise
    Commented May 1, 2021 at 15:23

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