Finding the Area Under a Graph

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 \[ A= \int\nolimits_{-1}^{1}e^{x}\, dx=\Big[e^{x}\Big] _{-1}^{1}=e^{1}-e^{-1} = e-\dfrac{1}{e} \approx 2.35. \]