Question

Use the Simpson’s rule withto estimate the area under the curvefromto

Short answer

The under estimate the area is 5.34 miles.

Step-by-step solution

solution Applied the interval from (1,4)

r (t) on the interval [1,4]

\(\begin{aligned}{l}d = \int\limits_1^4 {\frac{{{e^x}}}{x}dt} \\\Delta t = \frac{{b - a}}{n}\\ = \frac{3}{6}\\ = 0.5\end{aligned}\

Step 2: Solution of estimate the area

Use the Simpson’s Rule

\({S_n} = \frac{{\Delta t}}{3}\left( {f\left( {{t_0}} \right) + 4f\left( {{t_1}} \right) + 2f\left( {{t_2}} \right) + \cdots + f\left( {{t_n}} \right)} \right)\)

\(\begin{aligned}{l}{S_{10}} = \frac{{0.3}}{3}\left( {f\left( {\frac{1}{0}} \right) + 4f\left( {\frac{1}{{0.3}}} \right) + 2f\left( {\frac{1}{{0.6}}} \right) + \cdots + f\left( {\frac{1}{4}} \right)} \right)\\ \approx 5.34\end{aligned}\)