Chapter 5.9, Problem 29E

To determine

To find:

The total value of the given data of the Simpson’s rule.

Answer to Problem 29E

  ${\displaystyle \underset{0}{\overset{6}{\int }}\mathrm{a}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}}=37.37\mathrm{f}\mathrm{t}/\mathrm{s}$

Explanation of Solution

Given:

The graph of the data

Concept used:

Simpson’s formula for $\mathrm{n}$

  ${\mathrm{S}}_{10}=\frac{\mathrm{\Delta }\mathrm{x}}{3}\left[\begin{array}{l}\mathrm{f}\left({\mathrm{x}}_0\right)+4\mathrm{f}\left({\mathrm{x}}_1\right)+2\mathrm{f}\left({\mathrm{x}}_2\right)+4\mathrm{f}\left({\mathrm{x}}_3\right)+2\mathrm{f}\left({\mathrm{x}}_4\right)+4\mathrm{f}\left({\mathrm{x}}_5\right)+2\mathrm{f}\left({\mathrm{x}}_6\right)+4\mathrm{f}\left({\mathrm{x}}_7\right)\\ +2\mathrm{f}\left({\mathrm{x}}_8\right)+4\mathrm{f}\left({\mathrm{x}}_9\right)+\mathrm{f}\left({\mathrm{x}}_{10}\right)\end{array}\right]$

Calculation:

  $\begin{array}{l}\mathrm{n}=6,\mathrm{\Delta }\mathrm{t}=1\\ \mathrm{n}+1=7\text{terms}\end{array}$

The Simpson’s rule for $\mathrm{n}+1=7$ terms

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{6}{\int }}\mathrm{a}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}}=\frac{\mathrm{\Delta }\mathrm{t}}{3}\left[\mathrm{f}\left({\mathrm{t}}_0\right)+4\mathrm{f}\left({\mathrm{t}}_1\right)+2\mathrm{f}\left({\mathrm{t}}_2\right)+4\mathrm{f}\left({\mathrm{t}}_3\right)+2\mathrm{f}\left({\mathrm{t}}_4\right)+4\mathrm{f}\left({\mathrm{t}}_5\right)+\mathrm{f}\left({\mathrm{t}}_6\right)\right]\\ {\displaystyle \underset{0}{\overset{6}{\int }}\mathrm{a}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}}=\frac{1}{3}\left[0+4\left(0.5\right)+2\left(4.1\right)+4\left(9.8\right)+2\left(12.9\right)+4\left(9.5\right)+0\right]\\ {\displaystyle \underset{0}{\overset{6}{\int }}\mathrm{a}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}}=37.37\mathrm{f}\mathrm{t}/\mathrm{s}\end{array}$