Question

The area bounded by the curves $y=\sqrt{x},2y+3=x$ and $x$-axis in the $1^{st}$ quadrant is

• A. $9$
• B. $\frac{27}{4}$
• C. $36$
• D. $18$

Answer 1

The correct option is A $9$

Given curves are $y=\sqrt{x},2y+3=x$

$\Rightarrow y^2=2y+3\Rightarrow y=3,-1$

Then, the curves intersection in the first quadrant at $y=3,x=9$

To find the area between the curves $y=\sqrt{x},2y+3=x$ and $x$-axis in the $1^{st}$ quadrant
The required area is given by shaded portion $OABO$
$={\int }_0^9\sqrt{x}dx-{\int }_3^9\left(\frac{x-3}{2}\right)dx$

$={\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right)}_0^9-\frac{1}{2}{(\frac{x^2}{2}-3x)}_3^9$
$=\left(\frac{2}{3}.27\right)-\frac{1}{2}\{(\frac{81}{2}-27)-(\frac{9}{2}-9)\}$

$=18-\frac{1}{2}\left(18\right)=9$