A) \[\frac{56}{3}\]
B) \[\frac{60}{3}\]
C) \[\frac{66}{3}\]
D) \[\frac{62}{3}\]
Correct Answer: D
Solution :
Given, function is \[f(x)=\left\{ \begin{matrix} |x-2|+2, & x\le 2 \\ {{x}^{2}}-2, & x>2 \\ \end{matrix} \right.=\left\{ \begin{matrix} -(x-2)+2, & 0<x\le 2 \\ {{x}^{2}}-2, & x>2 \\ \end{matrix} \right.\] Now, \[\int_{0}^{4}{f(x)\,dx=\int_{0}^{2}{[-(x-2)+2]\,dx+\int_{2}^{4}{({{x}^{2}}-2)dx}}}\] \[=\int_{0}^{2}{(4-x)\,dx+\int_{2}^{4}{({{x}^{2}}-2)\,dx}}\] \[=\left[ 4x-\frac{{{x}^{2}}}{2} \right]_{0}^{2}+\left[ \frac{{{x}^{3}}}{3}-2x \right]_{2}^{4}\] \[=[8-2]+\left[ \frac{64}{3}-8-\frac{8}{3}+4 \right]\] \[=6+\frac{56}{3}-4=2+\frac{56}{3}=\frac{62}{3}\]
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