The area (in sq. units) of the region bounded

Question:

The area (in sq. units) of the region bounded by the curves $y=2^{x}$ and $y=|x+1|$, in the first quadrant is:

  1. $\frac{3}{2}-\frac{1}{\log _{\mathrm{e}} 2}$

  2. $\frac{1}{2}$

  3. $\log _{\mathrm{e}} 2+\frac{3}{2}$

  4. $\frac{3}{2}$


Correct Option: 1

Solution:

Required Area

$\int_{0}^{1}\left((x+1)-2^{x}\right) d x$

$=\left(\frac{x^{2}}{2}+x-\frac{2^{x}}{\ln 2}\right)_{0}^{1}$

$=\left(\frac{1}{2}+1-\frac{2}{\ln 2}\right)-\left(0+0-\frac{1}{\ln 2}\right)$

$=\frac{3}{2}-\frac{1}{\ln 2}$

Leave a comment