Question

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} $

Answer 1

Answer: (B)

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} $