Chapter 7 - Integration - 7.6 Numerical Integration - 7.6 Exercises - Page 413: 8

$$ \int_{2}^{4} \frac{1}{x^{3}} d x $$ Here $a=2, b=4,$ and $n=4,$ with $(b-a) / n=(4-2) / 4=1/2 $ as the altitude of each trapezoid. Then $x_{0}=2, x_{1}=2.5, x_{2}=3, x_{3}=3.5,$ and $x_{4}=4 .$ Now find the corresponding function values. The work can be organized into a table, as follows. $$ \begin{aligned} &n=4, b=4, a=2, f(x)=\frac{1}{x^{3}}\\ &\begin{array}{c|l|l} \hline i & x_{i} & f\left(x_{i}\right) \\ \hline 0 & 2 & 0.125 \\ 1 & 2.5 & 0.064 \\ 2 & 3 & 0.03703 \\ 3 & 3.5 & 0.02332 \\ 4 & 4 & 0.015625 \end{array} \end{aligned} $$ (a) the trapezoidal rule: $$ \begin{aligned} \int_{2}^{4} \frac{d x}{x^{3}} & \approx \frac{4-2}{4}\left[\frac{1}{2}(0.125)+0.064+0.03703\right.\\ &\left. \quad +0.02332+\frac{1}{2}(0.015625)\right] \\ & \approx \frac{1}{2}(0.19466) \approx 0.0973 \end{aligned} $$ (b) Simpson’s Rule: $$ \begin{aligned} \int_{2}^{4} \frac{d x}{x^{3}} & \approx \frac{4-2}{3(4)}[0.125+4(0.064)+2(0.03703)\\ & \quad +4(0.02332)+0.015625] \\ & \approx \frac{1}{6}(0.056397) \\ & \approx 0.0940 \end{aligned} $$ (c) Exact value: $$ \begin{aligned} \int_{2}^{4} \frac{d x}{x^{3}} &=\int_{2}^{4} x^{-3} d x=\left.\frac{x^{-2}}{-2}\right|_{2} ^{4}=\left.\frac{-1}{2 x^{2}}\right|_{2} ^{4} \\ &=\frac{-1}{32}+\frac{1}{8}\\ &=\frac{3}{32}\\ &=0.09375 \end{aligned} $$

$$ \int_{2}^{4} \frac{1}{x^{3}} d x $$ Here $a=2, b=4,$ and $n=4,$ with $(b-a) / n=(4-2) / 4=1/2 $ as the altitude of each trapezoid. Then $x_{0}=2, x_{1}=2.5, x_{2}=3, x_{3}=3.5,$ and $x_{4}=4 .$ Now find the corresponding function values. The work can be organized into a table, as follows. $$ \begin{aligned} &n=4, b=4, a=2, f(x)=\frac{1}{x^{3}}\\ &\begin{array}{c|l|l} \hline i & x_{i} & f\left(x_{i}\right) \\ \hline 0 & 2 & 0.125 \\ 1 & 2.5 & 0.064 \\ 2 & 3 & 0.03703 \\ 3 & 3.5 & 0.02332 \\ 4 & 4 & 0.015625 \end{array} \end{aligned} $$ (a) the trapezoidal rule: $$ \begin{aligned} \int_{2}^{4} \frac{d x}{x^{3}} & \approx \frac{4-2}{4}\left[\frac{1}{2}(0.125)+0.064+0.03703\right.\\ &\left. \quad +0.02332+\frac{1}{2}(0.015625)\right] \\ & \approx \frac{1}{2}(0.19466) \approx 0.0973 \end{aligned} $$ (b) Simpson’s Rule: $$ \begin{aligned} \int_{2}^{4} \frac{d x}{x^{3}} & \approx \frac{4-2}{3(4)}[0.125+4(0.064)+2(0.03703)\\ & \quad +4(0.02332)+0.015625] \\ & \approx \frac{1}{6}(0.056397) \\ & \approx 0.0940 \end{aligned} $$ (c) Exact value: $$ \begin{aligned} \int_{2}^{4} \frac{d x}{x^{3}} &=\int_{2}^{4} x^{-3} d x=\left.\frac{x^{-2}}{-2}\right|_{2} ^{4}=\left.\frac{-1}{2 x^{2}}\right|_{2} ^{4} \\ &=\frac{-1}{32}+\frac{1}{8}\\ &=\frac{3}{32}\\ &=0.09375 \end{aligned} $$