Find the area of the region enclosed by the graphs of $f(x) =e^{x}$ and $g(x) =\sqrt{x}$ and the lines $x=0$ and $x=1$.

Solution We begin by graphing the two functions and identifying the area $A$ to be found. See Figure 3.

Figure 3

From the graph, we see that $f(x) \geq g(x)$ on the interval $[0,1]$. Then, using the definition of area, we have $$\begin{eqnarray*} A &=&\int_{a}^{b}[f(x)-g(x)]~{\it dx}=\int_{0}^{1}( e^{x}-\sqrt{x}) ~{\it dx}\\[2pt] &=&\int_{0}^{1}e^{x}~{\it dx}-\int_{0}^{1}x^{1/2}~{\it dx}=\big[ e^{x}\big] _{0}^{1}-\left[ \dfrac{x^{3/2}}{\dfrac{3}{2}}\right] _{0}^{1} \\[5pt] &=&( e^{1}-e^{0}) -\dfrac{2}{3}( 1-0) =e-1-\dfrac{2}{3}=e-\dfrac{5}{3} \hbox{ square units} \end{eqnarray*}$$