If [x] denotes the greatest integer less than | vedclass

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(B) Let $I = \int_{0}^{2} x^{2}[x] d x$. Since $[x]$ is the greatest integer function, we split the integral at the integer points in the interval $[0

Vedclass · Accepted Answer

$\frac{7}{3}$

Vedclass · Answer

(B) Let $I = \int_{0}^{2} x^{2}[x] d x$. Since $[x]$ is the greatest integer function, we split the integral at the integer points in the interval $[0, 2]$. $I = \int_{0}^{1} x^{2}[x] d x + \int_{1}^{2} x^{2}[x] d x$. For $0 \le x For $1 \le x Substituting these values, we get: $I = \int_{0}^{1} x^{2}(0) d x + \int_{1}^{2} x^{2}(1) d x$. $I = 0 + \int_{1}^{2} x^{2} d x$. $I = \left[ \frac{x^{3}}{3} \right]_{1}^{2}$. $I = \frac{2^{3}}{3} - \frac{1^{3}}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$.

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

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