Q99 / 155 · WBJEE 2016 · Mathematics

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int_{0}^{2} x^{2}[x] d x$ equals

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

Solution

Let. $$ \begin{aligned} I &=\int_{0}^{2} x^{2}[x] d x \\ &=\int_{0}^{1} x^{2} \cdot 0 d x+\int_{1}^{2} x^{2} \times 1 d x \\ &=\left[\frac{x^{3}}{3}\right]_{1}^{2}=\left[\frac{8}{3}-\frac{1}{3}\right]=\frac{7}{3} \end{aligned} $$

Hence, the correct answer is option: 'B'.

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