Find the area under the graph of $f(x) =e^{x}$ from $-1$ to $1.$

Solution Figure 23 shows the graph of $f(x)=e^{x}$ on the closed interval $[-1,1]$.

The area $A$ under the graph of $f$ from $-1$ to $1$ is given by $$A= \int\nolimits_{-1}^{1}e^{x}\, dx=\Big[e^{x}\Big] _{-1}^{1}=e^{1}-e^{-1} = e-\dfrac{1}{e} \approx 2.35.$$