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A Scoop of Blancmange

Problem 226 Published on Friday, 2nd January 2009, 09:00 pm; Solved by 1471;
Difficulty rating: 65%

The blancmange curve is the set of points (x,y) such that 0 ≤ x ≤ 1 and ,
where s(x) = the distance from x to the nearest integer.

The area under the blancmange curve is equal to ½, shown in pink in the diagram below.

blancmange curve

Let C be the circle with centre (¼,½) and radius ¼, shown in black in the diagram.

What area under the blancmange curve is enclosed by C?
Give your answer rounded to eight decimal places in the form 0.abcdefgh