Question

The value of $\stackrel{11}{\underset{0}{\int }}[x{]}^3\cdot dx$ where $[.]$ denotes the greatest integer function, is

• A. $0$
• B. $14400$
• C. $2200$
• D. $3025$

Answer 1

The correct option is D $3025$

${\int }_0^{11}[x{]}^{3}dx$

$\left[x\right]=\{\begin{array}{cc}0& 0\le x<1\\ 1& 1\le x<2\\ 2& 2\le x<3\\ 9& 9\le x<10\\ 10& 10\le x<11\end{array}$

${\int }_0^{11}[x{]}^{3}dx={\int }_0^1[x{]}^{3}dx+{\int }_1^2[x{]}^{3}dx+....{\int }_{10}^{11}[x{]}^{3}dx$

$={\int }_0^{1}0dx+{\int }_1^2{1}^{3}dx+{\int }_2^3{2}^{3}dx+.{\int }_{10}^{11}{10}^{3}dx$

$=1^3(2-1)+2^3(3-2)+3^3(4-3)+...{10}^3(11-10)$

$=1^3+2^3+3^3+....{10}^3$

$=\frac{{10}^2(10+1{)}^2}{4}(\sum n^3=\frac{n^2(n+1{)}^2}{4})$

$=5^2(11{)}^2\Rightarrow 25\times 121\Rightarrow 3025$