The area bounded by the curve y2(2a–x) = x3 and the line x = 2a is _

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The area bounded by the curve y²(2a–x) = x³ and the line x = 2a is __ sq.units

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${πa}^{2}$

$\frac{3 {πa}^{2}}{4}$

$3 {πa}^{2}$

$\frac{3 {πa}^{2}}{2}$

answer is A.

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Detailed Solution

$\begin{aligned} y^{2} = \frac{x^{3}}{2 a - x} \\ y = \sqrt{\frac{x^{3}}{2 a - x}} \end{aligned} s y m m e t r i c a b o u t x a x i s \begin{aligned} Area = 2 \int_{0}^{2 a} \sqrt{\frac{x^{3}}{2 a - x}} d x \\ x = 2 a \sin^{2} θ \\ d x = 4 a \sin θ \cos θ d θ \end{aligned}$

$2 \int_{0}^{π / 2} \frac{{(2 {asin}^{2} θ)}^{3 / 2}}{\sqrt{2 a - 2 {asin}^{2} θ}} 4 asin θcos θdθ$

$- 2 \frac{(2 a) \sqrt{2 a}}{\sqrt{2 a}} \times 4 a \int_{0}^{π / 2} \frac{\sin^{3} θ}{\cos θ} \sin θ \cos θ d θ$

$\begin{aligned} = 16 a^{2} \int_{0}^{\frac{π}{2}} \sin^{4} θ d θ \\ = 16 a^{2} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{π}{2} \\ = 3 π a^{2} \end{aligned}$