Q. The area of the region bounded by the straight lines $x = 0$ and $x = 2$ and the curves $y = 2^{x}$ and $y = 2x - x^{2}$ is equal to
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) $\frac{4}{\log 2} - \frac{3}{2}$
Explanation
(b)
Required area
$= \int_{0}^{2}{\left\lbrack 2^{x} - \left( 2x - 2^{2} \right) \right\rbrack dx}$
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$= \left\lbrack \frac{2^{x}}{\log 2} - x^{2} + \frac{x^{3}}{3} \right\rbrack_{0}^{2}$
$= \frac{4}{\log 2} - 4 + \frac{8}{3} - \frac{1}{\log{2\ }}$
$= \left( \frac{3}{\log 2} - \frac{4}{3} \right)sq\ unit$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.