Question

If $\left[x\right]$ denotes the greatest integer less than or equal to $x$,

then the value of the integral ${\int }_0^2{x}^2\left[x\right]dx$ equals?

• A. $\frac{5}{3}$
• B. $\frac{7}{3}$
• C. $\frac{8}{3}$
• D. $\frac{4}{3}$

Answer 1

Answer: (B)

$\frac{7}{3}$

${\int }_0^2{x}^2\left[x\right]dx={\int }_0^1{x}^2\left[x\right]dx+{\int }_1^2{x}^2\left[x\right]dx$

$=0+{\int }_1^2{x}^{2}dx={\left[\frac{x^3}{3}\right]}_1^2=\frac{8}{3}-\frac{1}{3}=\frac{7}{3}$

Note in interval $[0,1),\left[x\right]=0$ and in $[1,2),\left[x\right]=1$