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.

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

- 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)=x*². (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.

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.

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