Question

The area bounded by the two branches of curve $(y-x{)}^2=x^3$ and the straight line $x=1$ is

• A. $1/5$ sq. units
• B. $3/5$ sq. units
• C. $4/5$ sq. units
• D. $8/4$ sq. units

Answer 1

Answer: (C)

$(y-x{)}^2=x^3$, where $x\ge 0$
$\Rightarrow y-x=\pm x^{3/2}$
or $y=x+x^{3/2}\left(1\right)$
$y=x-x^{3/2}\left(2\right)$
Function $\left(1\right)$ is an increasing function
Function $\left(2\right)$ meets $x$-axis, when $x-x^{3/2}=0$ or $x=0,1$
Also, for $0<x<1,x-x^{3/2}>0$ and for $x>1,x-x^{3/2}<0$
When $x\to \infty ,x-x^{3/2}\to -\infty$
From these information, we can plot the graph as shown

Required area $=\underset{0}{\overset{1}{\int }}\left[\left(x+x^{3/2}\right)-\left(x-x^{3/2}\right)\right]dx$
$=2\underset{0}{\overset{1}{\int }}{x}^{3/2}dx$
$=2{\left[\frac{x^{5/2}}{5/2}\right]}_0^1$
$=\frac{4}{5}$ sq. units