0 1 . 5 [ x 2 ] ? d x , where [ · ] denotes the greatest integer function, equals:

Question

$\int_{0}^{1.5} [x^{2}] d x,$ where $[\cdot]$ denotes the greatest integer function, equals:

Smart Achievers · Accepted Answer

(2)

Ans. $2 - \sqrt{2}$

Explanation :

$\int_{0}^{1.5} [x^{2}] d x = \int_{0}^{1} [x^{2}] d x + \int_{1}^{\sqrt{2}} [x^{2}] d x + \int_{\sqrt{2}}^{1.5} [x^{2}] d x$

$= 0 + \int_{1}^{\sqrt{2}} 1 d x + \int_{\sqrt{2}}^{1.5} 2 d x$

$= {[x]}_{1}^{\sqrt{2}} + 2 {[x]}_{\sqrt{2}}^{1.5}$

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

$= 2 - \sqrt{2}$

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