Question

The area enclosed between the curve $y^2(2a-x)=x^3$ and the line $x=2$ above the $x$-axis is

• A. $\pi a^2$ sq. units
• B. $\frac{3\pi a^2}{2}$ sq. units
• C. $2\pi a^2$ sq. units
• D. $3\pi a^2$ sq. units

Answer 1

Answer: (B)

The required area $A=\underset{0}{\overset{2a}{\int }}\sqrt{\frac{x^3}{2a-x}}dx$

put $x=2asi{n}^2\theta$
$\Rightarrow dx=2a2sin\theta cos\theta d\theta$
$\Rightarrow A=8a^2\underset{0}{\overset{\pi }{\int }}{\left(\frac{1-cos2\theta }{2}\right)}^{2}d\theta$
$=2a^2\underset{0}{\overset{\pi }{\int }}\left(1-2cos2\theta +co{s}^{2}2\theta \right)d\theta$
$=2a^2\underset{0}{\overset{\pi }{\int }}\left(1-2cos2\theta +\frac{1+cos4\theta }{2}\right)d\theta$
$=\frac{3\pi a^2}{2}$