Consider a circular strip of inner radius ‘x’ and outer radius ‘x+delta_x’.
Since the probability of a dart landing on any part of the dart board is the same,
the probability of the dart landing in that circular strip =
(area of the circular strip)/(area of the circle)
Numerator = Pi*(x+delta_x)^2 – Pi*x^2
=2*Pi*x*delta_x
/* we ignore the 2*Pi*delta_x^2 term because in the limit delta_x goes to 0 the term becomes insignificant */
therefore the probability of the dart landing in that circular strip = 2*x*delta_x/r^2
expected distance from center = Sum {(probability of dart landing in some location) * ( distance of location from center) } where the sum is over all points in the circle.
=
Limit Sum { (2*x*delta_x/r^2) * x}
delta_x -> 0

/*since all points on the circular strip are at distance x from the center (assuming delta_x is very small => “limit delta_x -> 0? is to be taken)
also sum is from x = 0 to x = r
*/
now Limit of a sum = definite integral
expected distance = (2/r^2) integral (x^2.dx)
where limits of integration are 0 to r
= 2/r^2 * (r^3/3)
= 2r/3