Question

Find the area of the region bounded by the following curves, the X-axis and the given lines: y = `sqrt(16 - x^2)`, x = 0, x = 4

Answer 1

Let A be the required area.
Consider the equation y = `sqrt(16 - x^2)`.

∴ A = `int_0^4 y*dx`

= `int_0^4 sqrt(16 - x^2)*dx`

= `int_0^4 sqrt((4)^2 - (x)^2)*dx`

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

= `[4/2 sqrt(16 - (4)^2) + (16)/(2)sin^-1 (4/4)] - [0/2 sqrt(16 - (0)^2) + (16)/(2) sin^-1(0/2)]`

= [2(0) + 8sin^–1 (1)] - [0 + 0]
= `8 xx pi/(2)`
∴ A = 4π q. units.