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The area in sq units of the region A = \left \{ \left ( x,y \right )=\left ( x-1 \right ) \left [ x \right ]\leq y\leq 2\sqrt{x},\: \: 0\leq x\leq 2\right \} where [t] denotes the greatest integer function is 
Option: 1 \frac{8}{3}\sqrt{2}-\frac{1}{2}
Option: 2 \frac{4}{3}\sqrt{2}+1
Option: 3 \frac{4}{3}\sqrt{2}-\frac{1}{2}
Option: 4 \frac{8}{3}\sqrt{2}-1

Answers (1)

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\\A=\{(x, y):(x-1)[x] \leqslant y \leqslant 2 \sqrt{x},\;\; 0 \leq x \leq 2\} \\\\\ =\{(x, y): 0 \leq y \leq 2 \sqrt{x}, \quad 0 \leq x<1;\;\; (x-1) \leq y \leq 2 \sqrt{x} ;\;\; 1 \leq x<2\}

\\=\int_{0}^{1} 2 \sqrt{x} d x+\int_{1}^{2}(2 \sqrt{x}-(x-1)) d x \\ =\frac{4}{3}+\left(\frac{8 \sqrt{2}}{5}-\frac{4}{3}-\frac{1}{2}\right) \\ =\frac{8 \sqrt{2}}{3}-\frac{1}{2}

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himanshu.meshram

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