Question

The area of the region bounded by the parabola $y^2=2x$ and the straight line $x-y=4$ is

• A. $18sq$ units
• B. 9 sq units
• C. 27 sq units
• D. 6 sq units

Answer 1

Answer: (A)

The intersecting points of the given curves are obtained by solving the equations $x-y=4$ and $y^2=2x$ for $x$ and $y$.
We have, $y^2=8+2$ y i.e., $(y-4)(y+2)=0$
which gives $y=4,-2$ and $x=8,2$.

Thus, the points of intersection are $(8,4)$ and $(2,-2)$.
$\therefore$ Area $=\underset{-2}{\overset{4}{\int }}\left(4+y-\frac{1}{2}{y}^2\right)dy$
$={\left|4y+\frac{y^2}{2}-\frac{1}{6}{y}^3\right|}_{-2}^4$
$=18\text{ sq units }$