question_answer

The area of the region bounded by the curves \[y={{2}^{x}}\] and \[y=2x-{{x}^{2}}\]between the ordinates \[x=0\]and \[x=2\]is

A)  \[\frac{2}{\log \,2}-\frac{4}{3}\]

B)  \[\frac{3}{\log \,2}-\frac{4}{3}\]

C)  \[\frac{1}{\log \,2}-\frac{4}{3}\]

D)  \[\frac{4}{\log \,2}-\frac{3}{2}\]

Correct Answer: B

Solution :

Required area \[=\left| \int_{0}^{2}{[{{2}^{x}}-2x+{{x}^{2}}]dx} \right|\] \[=\left| \left[ \frac{{{2}^{x}}}{\log \,2}-2.\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3} \right]_{0}^{2} \right|\] \[=\left| \frac{4}{\log \,2}-4+\frac{8}{3}-\frac{{{2}^{0}}}{\log \,2} \right|\] \[=\left| \frac{4}{\log \,2}-\frac{4}{3}-\frac{1}{\log \,2} \right|\] \[=\frac{3}{\log \,2}-\frac{4}{3}\]