mathschallenge.net

Problem

Four quarter circles are drawn from each vertex in a unit square.

Find the area of the shaded region.

Solution

Consider the diagram below.

By using the unit circle (see diagram below), we can find the area between 0.5 and a by integrating under the curve, x² + y² = 1.
That is, y = (1 x²); taking positive root, because we only want the top part of the circle.

Using the Pythagorean Theorem, 1² = a² + ½² a = 3/2.

Area under curve =

3/2 1/2

(1 x2) dx

Using the substitution, x = sin(u), dx = cos(u) du, and we get:

Area under curve	=		π/3 π/6	cos2(u) du
	=		π/3 π/6	½(1 sin(2u)) du

Area under curve

=

½[u + ½cos(2u)]

π/3 π/6

=

π/12

Area of rectangle = ½(3/2 1/2) = (3 1)/4.

Area of ¼ shaded region = π/12 (3 1)/4.

Hence the area of the shaded region is, π/3 3 + 1.