Question

Find the area of the region enclosed by the following curves $y_1=e^x$, $y_2=x^2-1,x=-1$ and $x=1.$

Answer 1

Area enclosed$=$Area of ABCDEF

Finding boundary conditions, $x_1=-1$ to $x_2=1$.

$\therefore {\int }_{x_1}^{x_2}$ABCDEF$=\int y_{1}dx-\int y_{2}dx$

$={\int }_{-1}^1{e}^{x}dx-{\int }_{-1}^1(x^2-1)dx$

$=[e^x{]}_{-1}^1-{[\frac{x^3}{3}-x]}_{-1}^1$

$\Rightarrow (e-\frac{1}{e})-[\frac{1}{3}-1-(-\frac{1}{3}+1)]$

$\Rightarrow e-\frac{1}{e}-(\frac{2}{3}-2)$

$\Rightarrow e-\frac{1}{e}+\frac{4}{3}$.