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$r$: cicle C1's radius

$w$,$h$: rectangle R1's edges: $x=w$, $y=h$, $x=0$, $y=0$

$(w>2r, h>2r)$

$S(x,y)$: area of intersection of C1 and R1 when center of C1 is at $(x, y)$

$X$: follows uniform distributed in $[0,w]$

$Y$: follows uniform distributed in $[0,h]$

$X$ and $Y$ is independent.

I want to know:

What is $E(S(X, Y))$, the expected value of area of intersection.

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I solved it by divide the rectangle region into 13 sub-regions.

enter image description here

$E(S(X,Y)) = \iint _{Rect} P(\sigma)S(x,y) d\sigma=\frac{1}{W \cdot H}\lbrack 4\iint_AS(x,y)d\sigma+2\iint_BS(x,y)+2\iint_CS(x,y)d\sigma+\iint_DS(x,y)d\sigma+4\iint_ES(x,y)d\sigma \rbrack$

Each sub-regions can be worked out with some line integral or some triangular function. And can be solved with Matlab.

Finally:

$E(S(X,Y))=\frac{\frac{R^4\, \left(3\, \pi + 11\right)}{3} + 2\, \left(H - 2\, R\right)\, \left(\pi\, R^3 - \frac{2\, R^3}{3}\right) + 4\, R^4\, \left(\pi - \frac{4}{3}\right) - 2\, \left(2\, R - W\right)\, \left(\pi\, R^3 - \frac{2\, R^3}{3}\right) - \pi\, R^2\, \left(H - 2\, R\right)\, \left(2\, R - W\right)}{H\, W}$

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