Conclusion

We have seen that in situations where it is impossible to know the function governing some phenomenon exactly, it is still possible to derive a reasonable estimate for the integral of the function based on data points. The idea is to choose a model function going through the data points and integrate the model function. The definition of an integral as a limit of Reimann sums shows that if you chose enough data points, the integral of the model function converges to the integral of the unkown function, so theoretically, numerical integration is on solid ground.

We have also seen that there are many practical factors that influence how well numerical integration works. Simple model functions may not emulate the behavior of the unkown function well. Complicated model functions are hard to work with. Problems with the number of data points, or the way in which the data was collected can have a major impact, and while we have explored some simple ways of estimating how accurate a particular numerical integral will be, this can be quite complicated in general.

Nonetheless, by using common sense, together with a solid grasp of what the integral means and how it is related to the geometry of the function being integrated, a creative scientist, mathematician or engineer can accomplish a great deal with numerical integration.

Do-It-Yourself!

All of the methods for modeling functions that we've considered so far correspond to fitting elementary functions through successive data points. We've looked at constant functions, linear functions, cubic functions and trigonometric functions. However, you are not limited to these options. In practice, you should use the type of model function that seems best for the job.

To conclude this introduction to numerical integration, we will consider what goes into coming up with a new kind of model function.

Question 15

One of the most elementary type functions not yet considered are quadratic polynomials.

Each quadratic function is completely determined by 3 points. So our quadratic model will string together parabolas end-to-end, each through 3 data points. That is, the first parabola passes through points 1,2,3; the second parabola passes through points 3,4,5, and so on. Verify using a sketch that we need an odd number of points if every point is going to belong to a parabola.
We can integrate under each parabola exactly, if we know the formula for the parabola. If the 3 points through which we are fitting have equally-spaced x-coordinates a, b, and c, the exact value of the integral over [a,c] is
h/3 ( f(a) + 4 f(b) + f(c) )
where h=b-a=c-b. Verify that this is true for the quadratic functions f(x)=1, f(x)=x, and f(x)=x². (Hint: b=(a+c)/2.) Argue why this must imply the result for arbitrary quadratic functions f(x)=Ax² + Bx + C.
Construct a numerical integration scheme based on the above result.

Question 16

For this entire lab, we have assumed that you are given experimental data points, and want to integrate an unknown function. There is another way that numerical integration is useful: to numerically approximate definite integrals of functions for which an explicit antiderivative does not exist.

Recall that exp(x) is the function "ee-to-the-x." In the study of statistics, it is necessary to solve integrals for which the integrand is exp(x²). Unfortunately, this function has no antiderivative in terms of elementary functions. The back of most statistics books, however, tabulate integrals for functions like this. This is accomplished by numerical integration.

For a taste of this, use the data below to obtain an approximation to the integral of exp(x²) over the interval [0,1].

      x         exp(x²)
    ----      ------------
      0.0        1.0
      0.1        1.01
      0.2        1.04
      0.3        1.09
      0.4        1.17
      0.5        1.28
      0.6        1.43
      0.7        1.63
      0.8        1.90
      0.9        2.25
      1.0        2.72

State explicitly what method you used to obtain the approximation.

Determine whether exp(x²) is monotone, and whether it is of constant concavity. Use this information to come up with an upper and lower bound for the true integral of exp(x²)over the interval [0,1].
If your calculator has a numerical integration feature built in, try to numerically integrate exp(x²) over the interval [0,1] and compare your answer to your calculator`s.

Question 17 (Bonus): Really Do-it-yourself

You may have been thinking to yourself, "If only they would do thus and such, they'd have a really good integration scheme." You may not have. In any event, come up with your own method of numerical integration! This could be some combination of methods we've already studied, or something completely different. Be creative! Is there some underlying model for the data that is associated with your method? Illustrate your method with pictures when appropriate, keeping in mind the pictures you've seen illustrating the other methods. What are the advantages and disadvantages to your scheme? For example, is the "error" small? Is the method easy to compute with?

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The Geometry Center Calculus Development Team

A portion of this lab is based on a problem appearing in the Harvard Consortium Calculus book, Hughes-Hallet, et al, 1994, p. 174

Last modified: Wed Jan 17 15:52:18 1996