## 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!

 H 1. $\int_{-1}^{1}\ (x^3+x^2)\ dx$ Left sum; $n = 4$ Right sum; $n = 4$ Trapezoid sum; $n = 4$ Simpson sum; $n = 4$ H 2. $\int_{0}^{1}\ (x^3 + 2x)\ dx$ Left sum; n = 4 Right sum; $n = 4$ Trapezoid sum; $n = 4$ Simpson sum; $n = 4$ H 3. $\int_{0}^{3}\ (x^2 + x^3)\ dx$ Left sum; $n = 6$ Right sum; $n = 6$ Trapezoid sum; $n = 6$ Simpson sum; $n = 6$ H 4. $\int_{0}^{1}\ \frac{4}{1 + x^2}dx$ 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.$\int_{0}^{1}\ \frac{4}{1 + x^2}dx$ Left sum; $n = 100$ Exact Answer: $\color{purple}{π} = 3.141592654...$ Right sum; $n = 100$ Trapezoid sum; $n = 100$ Simpson sum; $n = 100$ T 2.$\int_{0}^{1}\ e^{-x^2} dx$ 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. $\int_{0}^{2}\ 2x\ dx$ E 2. $\int_{0}^{3}\ x^3\ dx$ E 3. $\int_{1}^{5}\ \ln x\ dx$ E 4. $\int_{0}^{2}\ \sin x\ dx$

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. $\int_{0}^{2}\ x^4\ dx$     Trapezoid sum; error $≤ 0.0005$ N 2. $\int_{0}^{3}\ x^5\ dx$     Trapezoid sum; $3$ decimal places N 3. $\int_{0}^{2}\ x^4\ dx$     Simpson sum; error $≤ 0.0005$ E 4. $\int_{0}^{3}\ x^5\ dx$     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: junio, 2013