Question

The area (in sq. units) of the region $\left\{\right(x,y):x\ge 0,x+y\le 3,x^2\le 4y\text{and}y\le 1+\sqrt{x}\}$
is:

• A. $\frac{59}{12}$
• B. $\frac{3}{2}$
• C. $\frac{7}{3}$
• D. $\frac{5}{2}$

Answer 1

The correct option is D $\frac{5}{2}$


We can devide the whole section in two parts
$A_1=$ Area from $0\text{to}1$
$A_2=$ Area from $1\text{to}2$

Required area$=A_1+A_2$
$A_1=\underset{0}{\overset{1}{\int }}(1+\sqrt{x})dx-\underset{0}{\overset{1}{\int }}\left(\frac{x^2}{4}\right)dx{A}_2=\underset{1}{\overset{2}{\int }}\left(3-x\right)dx-\underset{1}{\overset{2}{\int }}\left(\frac{x^2}{4}\right)dx{A}_1+A_2=\frac{5}{2}\text{sq.units}$