Question

Sketch the region `{(x, 0) : y = sqrt(4 - x^2)}` and x-axis. Find the area of the region using integration.

Answer 1

Given that `{(x, 0) : y = sqrt(4 - x^2)}` 

⇒ y² = 4 – x²

⇒ x² + y² = 4 which is a circle.

Required area = `2 * int_0^2 sqrt(4 - x^2)  "d"x`

Since circle is symmetrical about y-axis

= `2 * int_0^2 sqrt((2)^2 - x^2)  "d"x`

= `2 * [x/2 sqrt(4 - x^2) + 4/2 sin^-1  x/2]_0^2`

= `2[(2/2 sqrt(4 - 4) + 2 sin^-1 (1)) - (0 + 0)]`

= `2[2 * pi/2]`

= 2π sq.units

Hence, the required area = 2π sq.units