Question

The area (in square units) of the region bounded by the $y-$axis and the curve $2x=y^2-1$ is:

• A. $\frac{\sqrt{2}}{3}$
• B. $2\sqrt{3}$
• C. $\frac{2}{3}$
• D. $2\sqrt{2}$

Answer 1

The correct option is C $\frac{2}{3}$

The given curve $2x=y^2-1\cdots \left(i\right)$
Puttig $x=0$ in equation $\left(i\right),$ we get
$y^2-1=0\Rightarrow y=\pm 1$
$\therefore$ The given curve intersects $y-$axis at the points $A(0,1)$ and $(0,-1)$

Required Area $=$ Area of shaded region $PABP$
$=2\times$ Area of region $PAOP$
[$\because$ the region is symmetrical about $x-$axis]
$=2\left|\underset{0}{\overset{1}{\int }}xdy\right|$
$[\because$ the curve lies on left of $y-$axis$]$
$=2\left|\underset{0}{\overset{1}{\int }}\frac{1}{2}(y^2-1)dy\right|$
$=\left|{[\frac{y^3}{3}-y]}_0^1\right|=|-\frac{2}{3}|=\frac{2}{3}$ sq. units.