Four quarter circles are drawn from each vertex in a unit square.
Find the area of the shaded region.
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 =|
|(1 x2) dx|
Using the substitution, x = sin(u), dx = cos(u) du, and we get:
|Area under curve||=|
|½(1 sin(2u)) du|
|Area under curve||=||½[u + ½cos(2u)]|
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.