Chapter 21. Integration of Functions

The task of finding the area enclosed by a given curve is an old problem. The ancient Egyptians, for example, tried to find the area of a circular field by relating it to the area of a square field. The search for a solution to the area problem gave rise to the field of integral calculus, where the integral of a function between two limits is equal to the area under that function. In modern science and engineering, integral equations are widely used to model physical phenomena including things such as aerodynamic analysis and radiation computations .

Integral equations can be divided into two types ”proper and improper. A proper integral is one with finite integration limits and whose function can be evaluated at every point along the range of integration. An improper integral is one that has a singularity at some point in the range of integration and/or has one or both integration limits equal to positive or negative infinity. In some cases you can recast an improper integral into a proper one by using a change of variables .

In this chapter we will discuss numerical techniques to solve both proper and improper integrals. In either case the algorithms involve deriving a polynomial expression that, when evaluated, approximates the integral value. The solution process involves dividing the integration range into subelements. Generally speaking, the more subelements used the greater the precision of the final result. Many of the numerical techniques are iterative, beginning with an initial division of the integration range and then refining the solution by increasing the number of subelements until the change in the solution is less than a specified error tolerance.

At the end of the chapter we will briefly discuss the more general families of integrals known as Fredholm and Volterra integrals. We will then use one of the solution methods developed earlier in the chapter to determine the lift and moment coefficient of a NACA 2412 airfoil according to thin airfoil theory. The specific topics we will discuss in this chapter are ”

• Trapezoidal algorithms

• Simpson's rule

• Solving improper integrals

• Gaussian quadrature formulas

• General integral types

• Example: thin airfoil theory