Question

The area include between the curve $y^2(2a-x)=x^3$ and its asymptote is

• A. $3\pi a$
• B. $3\pi a^2$
• C. $4\pi a^2$
• D. $6\pi a^2$

Answer 1

The correct option is B $3\pi a^2$


Area between the curve and the asymptote

$2\underset{0}{\overset{2a}{\int }}ydx=2\underset{0}{\overset{2a}{\int }}\sqrt{\left(\frac{x^3}{2a-x}\right)dx}$

$Put,x=2asi{n}^2\theta$
$sothat,dx=4asin\theta cos\theta d\theta$

$=2\underset{0}{\overset{\pi /2}{\int }}\sqrt{\frac{(2asi{n}^2\theta {)}^3}{2aco{s}^2\theta }}.4a\sin \theta \cos \theta d\theta$
$=16a^2\underset{0}{\overset{\pi /2}{\int }}si{n}^4\theta d\theta$
$=16a^2.\frac{3}{4}.\frac{1}{2}.\frac{\pi }{2}=3\pi a^2$