The Numerical Integration Problem

We have learned in Calculus how to compute the definite integral

where is a continuous function. If is the antiderivative of , then the Fundamental Theorem of Calculus states that

In most practical situations, however, the antiderivative is not available. For example, the Gaussian integral

cannot be evaluated analytically, since the antiderivative of is not an elementary function.

The purpose of this project is to compute numerically the definite integral ; in other words, we want an algorithm (that will translate in a piece of code) to compute an approximation of the integral, up to the desired accuracy.

In Calculus we learned that the integral is a limit of Riemann sums. On a computer, we also approximate the integral by a sum. We consider the set of node points

and approximate the integral by the sum

There are many possible, meaningfull choices for the node points values and the values of the weights ; different sets of values define different integration methods.

In what follows we focus on two simple algorithms: the trapezoidal and the Simpson method. All the methods discussed here use an equidistant set of node points. More precisely, if we define the step

the node points are