First slide

Evaluation of definite integrals

Question

$\int_{0}^{2} [x^{2}] dx, where [x] is the greatest$ $i n t e g e r \leq x i s$

Moderate

A

$5 + \sqrt{2} + \sqrt{3}$
B

$- 5 + \sqrt{2} - \sqrt{3}$
C

$5 - \sqrt{2} - \sqrt{3}$
D

$- 4 + \sqrt{3} - \sqrt{2}$

Solution

$\begin{array}{l} The function [x^{2}] varies as follows between x = (0, 2) \\ [x^{2}] = \{\begin{cases} 0 if 0 \leq x^{2} < 1, or 0 \leq x < 1 \\ 1 if 1 \leq x^{2} < 2 or 1 \leq x < \sqrt{2} \\ 2 if 2 \leq x^{2} < 3 or \sqrt{2} \leq x < \sqrt{3} \\ 3 if 3 \leq x^{2} < 4 or \sqrt{3} \leq x < 2 \end{cases} \\ \Rightarrow I = \int_{0}^{1} 0 . dx + \int_{1}^{\sqrt{2}} 1 . dx + \int_{\sqrt{2}}^{\sqrt{3}} 2 \cdot dx + \int_{\sqrt{3}}^{2} 3 dx \\ = 0 + (\sqrt{2} - 1) + 2 (\sqrt{3} - \sqrt{2}) + 3 (2 - \sqrt{3}) \\ = \sqrt{2} - 1 + 2 \sqrt{3} - 2 \sqrt{2} + 6 - 3 \sqrt{3} = 5 - \sqrt{2} - \sqrt{3} \end{array}$

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