Question

Find the area of the region bounded by y = `sqrt(4 − x^2)` and x axis using integration.

Answer 1

Limits of Integration:

The curve y = `sqrt(4 − x^2)` meets the x axis where y = 0

`sqrt(4 − x^2) = 0`

⇒ `x^2 = 4`

⇒ x = ± 2

The limits are from −2 to 2.

Area = `2 ∫_0^2 sqrt(2^2 − x^2)` dx

`∫ sqrt(a^2 − x^2) dx = x/2 sqrt(a^2 − x^2) + a^2/2 sin (−1)(x/a)`   ...[Using formula]

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

`Area = 2[(0 + 2 sin^(−1)(1)) − (0 + 0)]`

Area = `2 [2.π/2]`

Area = 2π sq. units