Subject: Calculus

Numerical Integration

Numerical integration can be considered as one of the many algorithms of numerical analysis. Numerical integration focuses on approximating the numerical value of a definite integral. There are two situations in which it is impossible to find the exact value of a definite integral.

The first situation arises from the fact that in order to evaluate using the Fundamental Theorem of Calculus we need to know an antiderivative of \int_a^bf(x)dx . Sometimes, however, it is difficult, or even impossible, to find an antiderivative. Take for example,

\int_0^1e^{x^2}dx

and

\int_{-1}^{1}\sqrt{1+x^3}dx

which are integrals that are impossible to calculate.

The second situation arises when the function is determined from a scientific experiment through instrument readings or collected data. There may be no formula for the function.

In both cases we need to find approximate values of definite integrals. We already know one such method. Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide \left [a, b\right ] into subintervals of equal length ∆x=\frac{\left (b-a\right )}{n} , then we have

\int_{a}^{b}f(x)dx\approx \sum_{i=1}^{n}f(x_{i}^{*})\Delta x

where x_i^* is any point in the i^{th} subinterval {x_i-1, x_i$}. If x_i^* is chosen to be the left endpoint of the interval, then x_i^*= x_i and we’ll have

\int_{a}^{b}f(x)dx\approx L_n \approx \sum_{i=1}^{n}f(x_{i-1})\Delta x

If f(x)≥0, then the integral represents an area and represents an approximation of this area by the rectangles shown in below.

If we choose to be the right endpoint, then and we have

\int_{a}^{b}f(x)dx\approx R_n \approx \sum_{i=1}^{n}f(x_{i}\Delta x

which can be represented as

These kinds of problems are often called as numerical quadrature, since it relates to the ancient problem of the quadrature of the circle, i.e., constructing a square with equal area to that of a circle. As is well known, even many relatively simple integrals cannot be expressed in finite terms of elementary functions, and thus must be evaluated by numerical methods. Even when a closed form analytical solution exists it may be preferable to use a numerical quadrature formula.

The following topics are widely used numerical integration methods.

Sub-topics in Numerical Integration
  1. Rectangle method
  2. Trapezium rule
  3. Simpson's rule
  4. Newton–Cotes formulas
  5. Gaussian quadrature