Question

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

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

Answer 1

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

Area of shaded region
$=\underset{0}{\overset{1}{\int }}(1+\sqrt{x})dx+\underset{1}{\overset{2}{\int }}(3-x)dx-\underset{0}{\overset{2}{\int }}\frac{x^2}{4}dx$
$=[x{]}_0^1+{\left[\frac{x^{\frac{3}{2}}}{3/2}\right]}_0^1+[3x{]}_1^2-{\left[\frac{x^2}{2}\right]}_1^2-{\left[\frac{x^3}{12}\right]}_0^2$
$=\frac{5}{2}$ sq. units.