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

Answer: (D)

$55$

Given that: ${\int }_0^{11}\left[x\right]dx$

$={\int }_0^1\left[x\right]dx+{\int }_1^2\left[x\right]dx+{\int }_2^3\left[x\right]dx+{\int }_3^4\left[x\right]dx+{\int }_4^5\left[x\right]dx+{\int }_5^6\left[x\right]dx+{\int }_6^7\left[x\right]dx$ $+{\int }_7^8\left[x\right]dx+{\int }_8^9\left[x\right]dx+{\int }_9^{10}\left[x\right]dx+{\int }_{10}^{11}\left[x\right]dx$

$={\int }_0^{1}0dx+{\int }_1^{2}1dx+{\int }_2^{3}2dx+{\int }_3^{4}3dx+{\int }_4^{5}4dx+{\int }_5^{6}5dx+{\int }_6^{7}6dx+{\int }_7^{8}7dx+{\int }_8^{9}8dx$ $+{\int }_9^{10}9dx+{\int }_{10}^{11}10dx$

$=0+1+2+3+4+5+6+7+8+9+10$

$=55$