question_answer

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

A)  9 sq unit             

B)  \[\frac{27}{4}\] sq unit

C)  36 sq unit           

D)  18 sq unit

Correct Answer: A

Solution :

Required area of shaded portion OABO \[=\int_{0}^{9}{\sqrt{x}}dx-\int_{3}^{9}{\left( \frac{x-3}{2} \right)dx}\] \[=\left[ \frac{{{x}^{3/2}}}{3/2} \right]_{0}^{9}-\frac{1}{2}\left[ \frac{{{x}^{2}}}{2}-3x \right]_{3}^{9}\] \[=\left( \frac{2}{3}.27 \right)-\frac{1}{2}\left[ \left( \frac{81}{2}-27 \right)-\left( \frac{9}{2}-9 \right) \right]\] \[=18-\frac{1}{2}(18)=9\]sq unit