Numerical implies something associated with numbers, and integration is finding a sum or total value. Numerical integration thus literally means the sum total of numbers.
In mathematics, numerical integration is the name given to numerical computation of integral when it becomes very difficult or infeasible or impossible to find its exact value. In other words, numerical integration is concerned with the computation of an approximate value for the definite integral rather than finding an exact value. Various methods of numerical are known. We shall be concerned with two methods, namely, the Trapezoidal Rule and Simpson’s Rule.
Definite Integral and Area Computation
The fundamental Theorem of Calculus gives us a method of computing the definite integral of a given continuous function f from a to b. The method is to find, by trial and error, an anti-derivative F of f and then to use the equation;
When the method works, it provides an exact value for the integral. However, the method succeeds only if the anti-derivative happens to be a function that can be described in a simple way. For many integrals, one cannot find a formula for the anti-derivative, and the method fails.
Such integrals can still be computed approximately using numerical integration. A computation of this kind becomes possible once we agree that a definite integral is nothing but a limit of the sum of rectangles or trapeziums constructed to cover the region bounded by the graph of the function f, two vertical lines x a and x = b and the x-axis.
Composite Trapezoidal Rule
One way of arriving at a more accurate numerical approximation of a definite integral is to break up the given interval [a, b] into some number n of sub-intervals, imply elementary trapezoidal rule to each interval, and then add up all the results. The use of trapezium gives rise to what is known as the trapezoidal rule. Here is how the trapezoidal rule goes:
Exercise – 20.1
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