Chapter 7 - Integration - Chapter Review - Review Exercises - Page 418: 69

$$ \int_{0}^{1} e^{x} \sqrt{e^{x}+4} d x $$ Here $a=0, b=1,$ and $n=4,$ with $(b-a) / n=(1-0) / 4=1/4 $ as the altitude of each trapezoid. Then $x_{0}=0, x_{1}=1/4 , x_{2}=1/2 , x_{3}=3/4 ,$ and $x_{4}=1 .$ Now find the corresponding function values. The work can be organized into a table, as follows. \begin{aligned} &\text { Trapezoidal Rule }\\ &n=4, b=1, a=0, f(x)=e^{x} \sqrt{e^{x}+4}\\ &\begin{array}{c|l|c} \hline i & x_{i} & f\left(x_{i}\right) \\ \hline 0 & 0 & 2.236 \\ 1 & 0.25 & 2.952 \\ 2 & 0.5 & 3.919 \\ 3 & 0.75 & 5.236 \\ 4 & 1 & 7.046 \\ \hline \end{array} \end{aligned} The trapezoidal rule: $$ \begin{aligned} \int_{0}^{1} e^{x} \sqrt{e^{x}+4} d x &=\frac{1-0}{4}\left[\frac{1}{2}(2.236)+2.952\right. \\ &\quad+3.919+5.236+\frac{1}{2}(7.046) \\ &\approx 4.187 \end{aligned} $$ Exact value: \begin{aligned} \int_{0}^{1} e^{x} \sqrt{e^{x}+4} d x &=\int_{0}^{1} \left(e^{x}+4\right)^{1 / 2} d (e^{x}) \\ &=\left.\frac{2}{3}\left(e^{x}+4\right)^{3 / 2}\right|_{0} ^{1} \\ &=\frac{2}{3}(e+4)^{3 / 2}-\frac{2}{3}(5)^{3 / 2} \\ & \approx 4.155 \end{aligned}

$$ \int_{0}^{1} e^{x} \sqrt{e^{x}+4} d x $$ Here $a=0, b=1,$ and $n=4,$ with $(b-a) / n=(1-0) / 4=1/4 $ as the altitude of each trapezoid. Then $x_{0}=0, x_{1}=1/4 , x_{2}=1/2 , x_{3}=3/4 ,$ and $x_{4}=1 .$ Now find the corresponding function values. The work can be organized into a table, as follows. \begin{aligned} &n=4, b=1, a=0, f(x)=e^{x} \sqrt{e^{x}+4}\\ &\begin{array}{c|l|c} \hline i & x_{i} & f\left(x_{i}\right) \\ \hline 0 & 0 & 2.236 \\ 1 & 0.25 & 2.952 \\ 2 & 0.5 & 3.919 \\ 3 & 0.75 & 5.236 \\ 4 & 1 & 7.046 \\ \hline \end{array} \end{aligned} The trapezoidal rule: $$ \begin{aligned} \int_{0}^{1} e^{x} \sqrt{e^{x}+4} d x &=\frac{1-0}{4}\left[\frac{1}{2}(2.236)+2.952\right. \\ &\quad+3.919+5.236+\frac{1}{2}(7.046) \\ &\approx 4.187 \end{aligned} $$ Exact value: \begin{aligned} \int_{0}^{1} e^{x} \sqrt{e^{x}+4} d x &=\int_{0}^{1} \left(e^{x}+4\right)^{1 / 2} d (e^{x}) \\ &=\left.\frac{2}{3}\left(e^{x}+4\right)^{3 / 2}\right|_{0} ^{1} \\ &=\frac{2}{3}(e+4)^{3 / 2}-\frac{2}{3}(5)^{3 / 2} \\ & \approx 4.155 \end{aligned}