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

Question

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) \(\frac{5}{3}\)
(B) \(\frac{7}{3}\)
(C) \(\frac{8}{3}\)
(D) \(\frac{4}{3}\)

Answer 1

Ans: (B)
\(: \int_{0}^{2} x^{2}[x] \cdot d x=\int_{0}^{1} x^{2} \times 0 d x+\int_{1}^{2} x^{2} \times 1 d x=\left(x^{3} / 3\right)_{1}^{2}=7 / 3\)