Technical Java: Applications for Science and Engineering

The general form of a simple integral equation is shown in Eq. (21.1).

Equation 21.1

Eq. (21.1) is an example of a larger class of integrals known as Fredholm integrals of the first kind. Fredholm integrals are discussed in more detail at the end of this chapter, but for the time being let's consider the simple integral shown in Eq. (21.1). The evaluation of the integral equation computes the area under the curve f ( x ) from x = a to x = b .

If the integration limits a and b are finite numbers and the function f ( x ) is nonsingular over the integration range, the integral is termed a proper integral. Proper integrals are usually solved using closed techniques, those that evaluate the function at the endpoints, f ( a ) and f ( b ). If either integration limit is or “ , or if there is a singularity in the function f ( x ) anywhere in the integration range, the integral is improper. An improper integral that evaluates to a finite value is convergent. Convergent improper integrals can be solved using what are known as open techniques that only evaluate f ( x ) between x = a and x = b . An improper integral that is infinite is a divergent integral. Obviously, divergent integrals cannot be solved because their values diverge.

Some integrals have precise, closed-form solutions. For those that do not, numerical methods have been developed to approximate the integral value. The numerical methods we will discuss in this chapter approximate the integral value by developing a polynomial expression based on values of the function being integrated at discrete locations along the range of integration. We will start by examining trapezoidal algorithms.