First slide

Evaluation of definite integrals

Question

The integral $I = \int_{0}^{2} [x^{2}] d x ([t]$ denotes the greatest
integer less than or equal to t) is equal to:

Moderate

A

$5 - 2 \sqrt{3}$
B

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

$6 - \sqrt{2} - \sqrt{3}$
D

$3 - \sqrt{2}$

Solution

$\begin{array}{l} \int_{0}^{2} [x^{2}] d x \\ = \int_{0}^{1} [x^{2}] d x + \int_{1}^{\sqrt{2}} [x^{2}] d x + \int_{\sqrt{2}}^{\sqrt{3}} [x^{2}] d x + \int_{\sqrt{3}}^{2} [x^{2}] d x \\ = \int_{0}^{1} 0 d x + \int_{1}^{\sqrt{2}} 1 d x + \int_{\sqrt{2}}^{\sqrt{3}} 2 d x + \int_{\sqrt{3}}^{2} 3 d x \\ = 0 + (\sqrt{2} - 1) + 2 (\sqrt{3} - \sqrt{2}) + 3 (2 - \sqrt{3}) \\ = - \sqrt{2} - \sqrt{3} + 5 \end{array}$

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