Numerical Integration
exercises to accompany
Calculus Applied to the Real World

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Approximate the following integrals using the stated sums. Note: You should do these calculations without using a special utility or program. Use the tabular method shown in the on-line text examples. Answers must be accurate to at least $5$ decimal places!

 
Use whatever technology you have at your disposal to approximate the following integrals using the stated sums.
Note: For these, answers must be accurate to at least $8$ decimal places!

In the following exercises, use the error estimate formulas in the on-line text to give upper bounds for the errors if the given integral is approximated by (a) a trapezoidal sum with $n = 10$ subdivisions (b)a Simpson sum with $n = 10$ subdivisions.
Note: Your bounds may not be the same as ours. Rule of thumb: Your bound is acceptable, provided it is equal to or greater than ours. The closer it is to ours, the more efficiently you have used the formula.

In the following exercises, use the error estimate formulas in the on-line text to give lower bounds for the number $n$ of subdivisions needed to approximate the given integral by the given kind of sum.
Note: Your bounds may not be the same as ours. Rule of thumb: Your bound is acceptable, provided it is equal to or greater than ours. The closer it is to ours, the more efficiently you have used the formula.

Communication and Reasoning Exercises

C 1. Looking at the error estimate for the trapezoid rule, by how much will the error shrink if you increase n by a factor of $10$? What does this say about the increase in the number of digits of accuracy of the estimate given by the rule?

C 2. Looking at the error estimate for Simpson's rule, by how much will the error shrink if you increase $n$ by a factor of $10$? What does this say about the increase in the number of digits of accuracy of the estimate given by the rule?

C 3. Name at least two kinds of functions for which the Simpson sum always gives the exact answer, but not the trapezoid sum. Explain.

C 3. For which kind of functions does the trapezoid sum, but not the Simpson sum, give the exact answer? Explain.


 

Return to Main Page
Hndex of On-Line Topics
Text for This Topic
Everything for Calculus
Utility: Numerical Integration Utility
TI-83: Graphing Calculator Programs

Last Updated: junio, 2013
Copyright © 1999 Stefan Waner & Steven R. Costenoble