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:

The box function is defined as the greatest integer function

$\left[x\right]=\left\{\begin{array}{l}\begin{array}{ccc}0& for& 0\le x<1\\ 1& for& 1\le x<2\\ 2& for& 2\le x<3\\ 3& for& 3\le x<4\\ 4& for& 4\le x<5\\ 5& for& 5\le x<6\end{array}\\ \begin{array}{ccc}6& for& 6\le x<7\\ 7& for& 7\le x<8\\ 8& for& 8\le x<9\\ 9& for& 9\le x<10\\ 10& for& 10\le x<11\end{array}\end{array}\right.$

Therefore from this information, the given integral can be split up into smaller integrals as

${\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$$...\left(i\right)$

$\left[x\right]$ is the greatest integer function, which gives the value of the greatest integer less than or equal to the number itself

Consider the interval $(0,1)$

For all $x\in \left(0,1\right)$ , $\left[x\right]=0$

Similarly consider the interval $(1,2)$

For all $x\in \left(1,2\right)$ , $\left[x\right]=1$ and so on

Using this $\left(i\right)$ can be written as

$\Rightarrow$${\int }_0^{11}\left[x\right]dx$$=$${\int }_0^{1}0dx+$${\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$

We know that $\int dx=x$

$\Rightarrow$${\int }_0^{11}\left[x\right]dx$$=$$0+{\left(x\right)}_1^2+{\left(x\right)}_2^3+{\left(x\right)}_3^4+{\left(x\right)}_4^5+{\left(x\right)}_5^6+{\left(x\right)}_6^7+{\left(x\right)}_7^8+{\left(x\right)}_8^9+{\left(x\right)}_9^{10}+{\left(x\right)}_{10}^{11}$

$\Rightarrow$${\int }_0^{11}\left[x\right]dx$$=$$0+{\left(x\right)}_1^2+2{\left(x\right)}_2^3+3{\left(x\right)}_3^4+4{\left(x\right)}_4^5+5{\left(x\right)}_5^6+6{\left(x\right)}_6^7+7{\left(x\right)}_7^8+8{\left(x\right)}_8^9+9{\left(x\right)}_9^{10}+10{\left(x\right)}_{10}^{11}$

$\Rightarrow$${\int }_0^{11}\left[x\right]dx$$=$$0+\left(2-1\right)+2\left(3-2\right)+3\left(4-3\right)+4\left(5-4\right)+5\left(6-5\right)+6\left(7-6\right)+7\left(8-7\right)+8\left(9-8\right)+9\left(10-9\right)+10\left(11-10\right)$

$\Rightarrow$${\int }_0^{11}\left[x\right]dx$$=$$0+1+2+3+4+5+6+7+8+9+10$

$\Rightarrow$${\int }_0^{11}\left[x\right]dx$$=$$55$

Hence, the value of the integral ${\int }_0^{11}\left[x\right]dx$ is $55$, so option, (D) is the correct answer.