Question

Area enclosed between the curve y²(4 - x) = x³ and line x = 4 above X-axis is ______.

Options

• 4π

• 6π

• 8π

• 12π

Answer 1

Area enclosed between the curve y²(4 - x) = x³ and line x = 4 above X-axis is 6π.

Explanation:

Required area = `int_0^4 "y" "dx" = int_0^4 sqrt(x^3/(4 - x))`dx

Put x = 4 sin² θ

⇒ dx = 8 sin θ cos θ dθ

∴ Required area = `int_0^(pi/2) (sqrt(64sin^6 theta)* 8 sin theta cos theta)/(sqrt(4 - 4 sin^2 theta)) "d"theta`

= `int_0^(pi/2) (4 * sin^3 theta * 8 sin theta cos theta)/(cos theta) "d"theta`

`= 32 `int_0^(pi/2) sin^4 theta "d" theta`

`= 32 * 3.1/4.2*pi/2 = 6pi`