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 $=\left|{\int }_0^2[2^x-2x+x^2]dx\right|$
$=\left|{\left[\frac{2^x}{\log 2}-2.\frac{x^2}{2}+\frac{x^3}{3}\right]}_0^2\right|$
$=\left|\frac{4}{\log 2}-4+\frac{8}{3}-\frac{2^0}{\log 2}\right|$
$=\left|\frac{4}{\log 2}-\frac{4}{3}-\frac{1}{\log 2}\right|$
$=\frac{3}{\log 2}-\frac{4}{3}$