Question

If $\left[x\right]$is the greatest integer function not greater than $x$, then${\int }_0^{11}\left[x\right]dx=$

• A. $45$
• B. $66$
• C. $35$
• D. $55$

Answer 1

The correct option is D

$55$

Explanation for the correct answer:

To find the value of ${\int }_0^{11}\left[x\right]dx$ :

Let us consider $y={\int }_0^{11}\left[x\right]dx$

Then,

$\begin{array}{rcl}& & \\ & & \begin{array}{l}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}y={\int }_0^{11}\left[x\right]dx\end{array}\end{array}\end{array}\\ ={\int }_0^1\left[x\right]dx+{\int }_1^2\left[x\right]dx+{\int }_2^3\left[x\right]dx\dots .+{\int }_{10}^{11}\left[x\right]dx\\ ={\int }_0^{1}0dx+{\int }_1^{2}1dx+{\int }_2^{3}2dx+\cdots +{\int }_{10}^{11}10dx\\ =0+[x{]}_1^2+[2x{]}_2^3+[3x{]}_3^4+\cdots +10[x{]}_{10}^{11}\\ =(2-1)+2(3-2)+3(4-3)+\cdots +10(11-10)\\ =1+2+3+\cdots +10\\ =10\times \frac{(10+1)}{2}\\ =\frac{10\times 11}{2}\\ =55\end{array}\end{array}\end{array}$

Hence, the correct option is D.