Answered: Use (a) the Trapezoidal Rule, (b) the…

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter65: Achievement Review—section Six

Section: Chapter Questions

Problem 44AR: Solve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places...

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Question

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule,
and (c) Simpson’s Rule n=10 with to approximate the given
integral. Round your answers to six decimal places.

∫₂⁴1/In x dx

Expert Solution

Step 1

$\begin{array}{l} (a) Given \\ \int_{2}^{4} \frac{1}{\ln (x)} d x, n = 10 \\ Using Trapezoidal Rule \\ \int_{a}^{b} f (x) d x \approx \frac{Δ x}{2} [f (x_{o}) + 2 f (x_{1}) + 2 f (x_{2}) + ...... + 2 f (x_{n - 1}) + f (x_{n})] \\ Where, \\ Δ x = \frac{b - a}{n} \\ Now \\ Δ x = \frac{4 - 2}{10} = \frac{1}{5} \\ Now the interval [2,4] divide into n=10 subintervals of length Δ x= \frac{1}{5} \\ a = 2, \frac{11}{5}, \frac{12}{5}, \frac{13}{5}, \frac{14}{5}, 3, \frac{16}{5}, \frac{17}{5}, \frac{18}{5}, \frac{19}{5}, 4 = b \\ Now put in formula \\ \int_{a}^{b} f (x) d x \approx \frac{Δ x}{2} [f (x_{o}) + 2 f (x_{1}) + 2 f (x_{2}) + ...... + 2 f (x_{n - 1}) + f (x_{n})] \\ \int_{2}^{4} \frac{1}{\ln (x)} d x \approx \frac{1}{10} [f (2) + 2 f (\frac{11}{5}) + 2 f (\frac{12}{5}) + 2 f (\frac{13}{5}) + 2 f (\frac{14}{5}) + 2 f (3) + 2 f (\frac{16}{5}) + 2 f (\frac{17}{5}) + 2 f (\frac{18}{5}) + 2 f (\frac{19}{5}) + f (4)] \\ \int_{2}^{4} \frac{1}{\ln (x)} d x \approx 1.925444 \end{array}$

Step 2

$\begin{array}{l} (b) Given \\ \int_{2}^{4} \frac{1}{\ln (x)} d x, n = 10 \\ Using Midpont Rule \\ \int_{a}^{b} f (x) d x \approx Δ x [f (\frac{x_{o} + x_{1}}{2}) + f (\frac{x_{1} + x_{2}}{2}) + f (\frac{x_{2} + x_{3}}{2}) + ...... + f (\frac{x_{n - 2} + x_{n - 1}}{2}) + f (\frac{x_{n - 1} + x_{n}}{2})] \\ Where, \\ Δ x = \frac{b - a}{n} \\ Now \\ Δ x = \frac{4 - 2}{10} = \frac{1}{5} \\ Now the interval [2,4] divide into n=10 subintervals of length Δ x= \frac{1}{5} \\ a = 2, \frac{11}{5}, \frac{12}{5}, \frac{13}{5}, \frac{14}{5}, 3, \frac{16}{5}, \frac{17}{5}, \frac{18}{5}, \frac{19}{5}, 4 = b \\ Now put in formula \\ \int_{a}^{b} f (x) d x \approx Δ x [f (\frac{x_{o} + x_{1}}{2}) + f (\frac{x_{1} + x_{2}}{2}) + f (\frac{x_{2} + x_{3}}{2}) + ...... + f (\frac{x_{n - 2} + x_{n - 1}}{2}) + f (\frac{x_{n - 1} + x_{n}}{2})] \\ \int_{2}^{4} \frac{1}{\ln (x)} d x \approx \frac{1}{5} [f (\frac{21}{10}) + f (\frac{23}{10}) + f (\frac{5}{2}) + f (\frac{27}{10}) + f (\frac{29}{10}) + f (\frac{31}{10}) + f (\frac{33}{10}) + f (\frac{7}{2}) + f (\frac{37}{10}) + f (\frac{39}{10})] \\ \int_{2}^{4} \frac{1}{\ln (x)} d x \approx 1.920915 \end{array}$

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