The area (in sq. units) of the region bounded by the curves y = 2 ^...

We know that \left|x\right| = \left\{\begin{cases}x, & x \ge 0 \\ - x, & x < 0\end{cases}\right)</p><p>Hence,\left|x + 1\right| = \left\{$\begin{cases}x + 1, & x \ge - 1 \\ - \left(x + 1\right), & x < - 1\end{cases}$\right)
The graph of the given functions is

The shaded area is the required area.

Area $= {∫}_{0}^{1}\left({y}_{line} - {y}_{curve}\right)dx$

Area $= {∫}_{0}^{1}\left(x + 1 - {2}^{x}\right)dx$

Using $∫ {x}^{n}dx = \dfrac{{x}^{n + 1}}{n + 1} \text{\text{ and } } ∫ {a}^{x}dx = \dfrac{{a}^{x}}{\log_{e}a},$ we get

Area $= {\left[\dfrac{{x}^{2}}{2} + x - \dfrac{{2}^{x}}{\log_{e}2}\right]}_{0}^{1}$

Area $= \left(\dfrac{1}{2} + 1 - \dfrac{2}{\log_{e}2}\right) - \left(0 + 0 - \dfrac{1}{\log_{e}2}\right)$

Area $= \dfrac{3}{2} - \dfrac{1}{\log_{e}2}$ sq units.