Question

The area bounded by $y=x^2,$ $y=[x+1],$ $x\le 1$ and the $y$-axis, where $[.]$ represents the greatest integer function, is

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{7}{3}$
• D. $1$

Answer 1

The correct option is A $\frac{2}{3}$

$y=x^2,y=[x+1]=\left[x\right]+1$
For $0\le x<1,y=0+1=1$


Hence, the required area $=$ the shaded area
$=\underset{0}{\overset{1}{\int }}xdy=\underset{0}{\overset{1}{\int }}\sqrt{y}dy$
$=\frac{2}{3}$