Question

Find the area bounded by the curve $y=\sqrt{x},x=2y+3$ in the first quadrant and x-axis.

Answer 1

Given equation of the curves are for $y=\sqrt{x}$ and $x=2y+3$ in the first quadrant.

On solving both the equation for $y$, we get

$y=\sqrt{2y+3}$

$\Rightarrow y^2=2y+3$

$\Rightarrow y^2-2y-3=0$

$\Rightarrow y^2-3y+y-3=0$

$\Rightarrow y(y-3)+1(y-3)=0$

$\Rightarrow (y+1)(y-3)=0$

$\Rightarrow y=-1,3$

$\therefore$ Required area of shaded region,

$A={\int }_0^3(2y+3-y^2)dy={[\frac{2y^2}{2}+3y-\frac{y^3}{3}]}_0^3$

$=[\frac{18}{2}+9-9-0]=9squnits$.