Question

1. Estimate the errors involved in Exercise 57, parts (a) and (b). How large should n be in each case to guarantee an error of less than 0.00001?

Short answer

1. The error estimation is |Es| < 0.01348
2. The large error size is n > 368

Step-by-step solution

Step 1: Applied Simpson’s Rule

By using Simpson’s Rule

\(\begin{aligned}{l}f(x) = \int\limits_2^4 {\frac{1}{{\ln x}}dx} \\\Delta x = \frac{{4 - 2}}{{10}}\\ = 0.5\\{S_{10}} = (f(0) + 2f(0.5) + 2f(0.5) + \cdots + 2f(2) + f(4))\frac{{\Delta x}}{3}\\ \approx 0.01348\end{aligned}\)

Step 2: Solution of  to approximate by using Simpson’s rule

Let, consider the interval, there are 10 subintervals to consider

The length of each interval

\(\begin{aligned}{l}\int\limits_2^4 {\frac{1}{{\ln x}}dx} \approx {S_{10}}\\ \approx 0.01348\end{aligned}\)

The error estimation is

Explanation of error size in using  is less than 0.00001

The errorby using Simpson’s rule\(\left| {{E_S}} \right| \le \frac{{K{{(b - a)}^5}}}{{180{n^4}}}\)

\(\begin{aligned}{l}\left| {{E_S}} \right| < 0.00001\\\frac{{3.8{{(4 - 2)}^5}}}{{180{n^4}}} < 0.00001\\{n^4} > \frac{{3.8{{(2)}^5}}}{{180(0.00001)}}\\n \ge 368\end{aligned}\)

The large error size is n > 368