First slide

Evaluation of definite integrals

Question

$\int_{0}^{3 / 2} [x^{2}] d x,$ where $[]$ denotes the greatest integer function is

Moderate

A

$2 + \sqrt{2}$
B

$2 - \sqrt{2}$
C

$4 + 2 \sqrt{2}$
D

$4 - 2 \sqrt{2}$

Solution

$\int_{0}^{3 / 2} [x^{2}] d x = \int_{0}^{1} [x^{2}] d x + \int_{1}^{\sqrt{2}} [x^{2}] d x + \int_{\sqrt{2}}^{3 / 2} [x^{2}] d x$
$= \int_{0}^{1} 0 d x + \int_{1}^{\sqrt{2}} 1 d x + \int_{\sqrt{2}}^{3 / 2} 2 d x$

$= 0 + {[x]}_{1}^{\sqrt{2}} + {[2 x]}_{\sqrt{2}}^{3 / 2}$

$= \sqrt{2} - 1 + 3 - 2 \sqrt{2} = 2 - \sqrt{2}$

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