Integral Trapezoidal Rule

Introduction to integral trapezoidal rule :

The integral trapezoidal rule is a method which has been used to calculate the definite integral that has been given by

a
∫ f(x) dx
b

A definite integral is defined as the integral which has both the upper and the lower limits. In this topic we are going to see about the integral trapezoidal rule. The integral trapezoidal rule is also called as the trapezium rule or the trapezoid rule.

Integral Trapezoidal Rule:

The trapezoidal rule has been used by considering the region that has been under the graph of the given function f(x) as a trapezoid and then finding its area.

a
∫ f(x) dx ≈ (b-a) f(a)+f(b)
b                        2

The next step is to split the interval of the integration (a, b) into a number of smaller intervals and then apply the trapezoidal rule on every subintervals. This is given by

a                                            n-1
∫ f(x) dx ≈ (b-a)   (f(a)+f(b))/2   + ∑ f(a+k((b-a)/n)
b                 n                         k=1

This integration can also be written as

a
∫ f(x) dx ≈ ((b-a)/2n)(f(x0)+2f(x1)+2f(x2)+…+2f(xn-1)+f(xn))
b

Where the value of xk = a + k(b-a)/n, for the values of k = 0, 1, …. N

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Error Analysis of the Integral Trapezoidal Rule:
The error of the integral trapezoidal rule is given by the difference of the integral and the result.

b                                                n-1     
Error = ∫ f(x) dx -  b-a   (f(a)+f(b)/2) + 2   + ∑ f(a+k((b-a)/n)
a                 n                             k=1

This can be simply written as

Error = - (b-a)3    f′′(ξ)
12n2

Integral trapezoidal rule for non uniform intervals:

Let us consider the given values are x1 ….xn and y1,…..yn, then according to the integral trapezoidal rule

n
∫ydx ≈ ½ ∑ (xi – xi-1)(yi+yi-1)
i=2

The integral trapezoidal rules are said to be exact when the periodic functions are said to be integrated with their periods and the trapezoidal rule converges very rapidly for the periodic functions. When the function is said to be periodic and it integrates over one complete period then there exists a number of graphs which makes the error of the trapezoidal rule to be cancelled.