Integral Calculus
86727
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
1 \(\frac{5}{3}\)
2 \(\frac{7}{3}\)
3 \(\frac{8}{3}\)
4 \(\frac{4}{3}\)
Explanation:
(B) : Given,
\(\int_{0}^{2} x^{2}[x] d x=\int_{0}^{1} x^{2}[x] d x+\int_{1}^{2} x^{2}[x] d x=\)
\(0+\int_{1}^{2} \mathrm{x}^{2} \mathrm{dx}\)
\(=\left[\frac{\mathrm{x}^{3}}{3}\right]_{1}^{2}=\frac{8}{3}-\frac{1}{3}=\frac{7}{3}\)