First slide

Evaluation of definite integrals

Question

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

Easy

A

$\frac{7}{3}$
B

$\frac{3}{7}$
C

$\frac{2}{7}$
D

$\frac{7}{2}$

Solution

$\begin{matrix} \int_{0}^{2} x^{2} [x] d x = \int_{0}^{1} x^{2} [x] d x + \int_{1}^{2} x^{2} [x] d x \\ = \int_{0}^{1} x^{2} (0) d x + \int_{1}^{2} x^{2} (1) d x \\ = 0 + \int_{1}^{2} x^{2} d x = {|\frac{x^{3}}{3}|}_{1}^{2} = \frac{8 - 1}{3} = \frac{7}{3} \end{matrix}$

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