## Quarter Circles

#### 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, `x`² + `y`^{2} = 1.

That is, `y` = (1 `x`^{2}); taking positive root, because we only want the top part of the circle.

Using the Pythagorean Theorem, 1^{2} = `a`^{2} + ½^{2} `a` = 3/2.

Area under curve = |
| (1 x^{2}) dx |

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

Area under curve | = |
| cos^{2}(u) du | |||

= |
| ½(1 sin(2u)) du |

Area under curve | = | ½[u + ½cos(2u)] |
| = | 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.

Problem ID: 142 (Dec 2003) Difficulty: 4 Star