Question

Area of the region bounded by y=$\sqrt{x},x=2y+3$ & x-axis lying in 1st quadrant is:

• A. 2$\sqrt{3}$
• B. 18
• C. 9
• D. 34/3

Answer 1

The correct option is C 9

$\begin{array}{c}Wehave\\ y=\sqrt{x}andx=2y+2\\ solvingweget\\ y=\sqrt{2}y+3,y\ge 0\\ \Rightarrow y^2=2y+3,y\ge 0\\ \Rightarrow y^2-2y-3=0,y\ge 0\\ \Rightarrow \left(y-3\right)\left(y+1\right)=0,y\ge 0\\ Thegraphoffunctiony=\sqrt{x}ispartofparabola{y}^2=xlyingabovex-axis.\\ Thegraphisasshowninthefigureinthefollowingfigure.\\ Fromthefigur,areaofshadedregion,\\ A={\int }_0^3\left(2y+3-y^2\right)dy\\ ={\left[\frac{2y^2}{2}+3y-\frac{y^3}{3}\right]}_0^3\\ =\left[\frac{18}{2}+9-9-0\right]\\ =9sq.units\end{array}$

Hence,the option $C$ is the correct answer.