## Numerical Integration exercises to accompany Calculus Applied to the Real World

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Everything for Calculus
Utility: Numerical Integration Utility
TI-83: Graphing Calculator Programs

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!

H 1.  1  -1 (x3+x2) dx
Left sum; n = 4
Right sum; n = 4
Trapezoid sum; n = 4
Simpson sum; n = 4
H 2.  1  0 (x3 + 2x) dx
Left sum; n = 4
Right sum; n = 4
Trapezoid sum; n = 4
Simpson sum; n = 4
H 3.  3  0 (x2 + x3) dx
Left sum; n = 6
Right sum; n = 6
Trapezoid sum; n = 6
Simpson sum; n = 6
H 4.  1  0 41 + x2 dx
Left sum; n = 6
Right sum; n = 6
Trapezoid sum; n = 6
Simpson sum; n = 6

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!

T 1.  1  0 41 + x2 dx
Left sum; n = 100
Exact Answer: = 3.141592654... Right sum; n = 100
Trapezoid sum; n = 100
Simpson sum; n = 100
T 2.  1  0 e-x2 dx
Left sum; n = 100
Exact Answer: 0.7468241328... Right sum; n = 100
Trapezoid sum; n = 100
Simpson sum; n = 100

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.

E 1.  2  0 2x dx
E 2.  3  0 x3 dx
E 3.  5  1 ln x dx
E 4.  2  0 sin x dx

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.

N 1.  2  0 x4 dx Trapezoid sum; error 0.0005
N 2.  3  0 x5 dx Trapezoid sum; 3 decimal places
N 3.  2  0 x4 dx Simpson sum; error 0.0005
E 4.  3  0 x5 dx Simpson sum; 3 decimal places

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.

Last Updated: September, 1999