SUNY Geneseo Department of Mathematics
Math 230 02
Spring 2017
Prof. Doug Baldwin
Complete by Sunday, April 9
Grade by Tuesday, April 11
This lesson introduces you to two classic algorithms for evaluating definite
integrals. In the process, it also develops your ability to program with
for loops, and reinforces your ability to work with vectors
and plotting.
Numerical integration is the process of evaluating a definite integral for its numeric value. A number of algorithms for numerical integration have been developed, starting in the days when people had to evaluate integrals by hand, and more recently intended for implementation on computers. The Wikipedia article on numerical integration provides a good sense of the variety of methods that are employed, although it covers too much material in too little space to give a very good understanding of any one method.
You will explore two classical simple numerical integration algorithms in this lesson: Riemann sums and the trapezoid rule. For an introduction to Riemann sums see the Khan Academy “Riemann Approximation Introduction” video at https://www.khanacademy.org/math/ap-calculus-ab/riemann-sums-ab/rieman-sums-tut-ab/v/simple-riemann-approximation-using-rectangles. For the trapezoid rule, see “Trapezoidal Riemann Approximations” at https://www.khanacademy.org/math/ap-calculus-ab/riemann-sums-ab/trapezoidal-rule-ab/v/rectangular-and-trapezoidal-riemann-approximations. We will discuss both of these algorithms, including developing pseudocode, in class on March 31.
This lesson also introduces a new Matlab feature, for loops. Our textbook
discusses for loops in section 12.4, and we will talk about them in class
on April 3.
Riemann sums and the trapezoid rule are both algorithms for evaluating definite integrals. Any time you have multiple algorithms for solving the same problem the question of which algorithm is “best” arises. The answer usually depends on context, i.e., sometimes you need a fast solution above all else, sometimes you need one that is very accurate, sometimes you need one that fits in a small amount of computer memory, and very often you need one that strikes a balance between many criteria. In this lesson you will compare Riemann sums and the trapezoid rule to see which one most quickly converges on a close approximation to the actual value of an integral. The comparison will be qualitative rather than quantitative, and, as you will see, not entirely clear cut.
Do the following.
Write two Matlab functions, one of which uses Riemann sums to compute a definite integral and the other of which uses the trapezoid rule.
Both functions should take a function handle for the function being integrated, the bounds of the interval over which to integrate, and the number of sub-intervals to divide that interval into as arguments. Both functions should return the estimated integral as their result.
Write new, extended, versions of your functions from Step 1, which, in addition to calculating the approximate numeric value of each integral, also draw a graph showing the function they are integrating and the rectangles or trapezoids that they use to approximate the area under it. The modified functions should still return the estimated integral.
I suggest that you write these as new functions rather than just changing the ones from Step 1 because you will want the original functions from Step 1 in Step 3.
Use these functions to visually explore how Riemann sums and the trapezoid rule approximate the area under the function y = x3 between x = 0 and x = 1. Try this for a number of sub-intervals (usually called n in discussions of numerical integration) ranging from 4 or 5 up to 100 or more. Then repeat this exploration for the function y = sinx between x = 0 and x = π. Based on what you see, which integration algorithm do you think will produce the most accurate results with the fewest sub-intervals?
To test your insights from Step 2, write a Matlab script that visualizes how close the estimated integral computed by each of the functions from Step 1 is to the actual integral. This script should step through a number of n values, call the Riemann and trapezoid functions for each n, and plot the returned values against n. Your script should also plot a straight line whose Y value is the actual value of the integral being estimated by the Riemann and trapezoid functions. (You can calculate this actual value by hand and make it a constant in your program, you don't have to write Matlab code to do it.)
Run your script for the function y = x3 with x ranging from 0 to 1. Also run your script for y = sinx with x ranging from 0 to π. Are the visualizations consistent with what you expected from inspecting plots in Step 2? In general, would you favor Riemann sums over the trapezoid rule for computing definite integrals, or vice versa, or can you not say? What caveats come with your answer?
The error in integrating f(x) from a to b by the trapezoid rule (i.e., the difference between the actual integral and the value found using the trapezoid rule) is bounded by the inequality
|E| ≤ M(b-a)3 / (12n2)
where E is the error and M is an upper bound on the absolute value of the second derivative of f in the interval [a, b]. As usual, n is the number of intervals used by the trapezoid rule.
For up to 2 points extra credit, extend the plots you draw in Step 3 to also plot the error envelope for the trapezoid rule, i.e., curves showing the actual value of the integral plus and minus the maximum error as calculated by the above relation. How large is this error envelope? Are the values produced by the trapezoid rule generally well inside it, or are they close to one edge of the envelope?
I will demonstrate the kinds of plots and functions I want you to write for this exercise in class.
I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring your solution on your computer, as that will speed the process along.
Sign up for a meeting via Google calendar. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.