Numerical Integration

Objective

Our goal is to estimate the value of a definite integral of the form

Definite integral

We divide the interval [a, b] into n equally sized subintervals where a = x0 < x1 < ⋅⋅⋅ < xn = b and where xi+1 = xi + Δx and Δx = (b–a)/n. Thus, xi = a + Δx⋅i.

Midpoint Rule

A commonly used way of estimating the definite interval is as follows:

Midpoint rule

Here, we assume

Definition of z_i

Thus

Iterative approach

The maximum error using this approach is

Maximum error midpoint rule

where

M_2

Trapezoid Rule

Another commonly used way of estimating the definite interval is as follows:

Trapezoid rule

The maximum error for the trapezoid rule is

Maximum error trapezoid rule

Simpson’s Rule

The final estimation approach that we will consider is

Simpson's rule

Here, n is even with n = 2m. The maximum error for Simpson’s rule is

Maximum error Simpson's rule

Non-finite limits

The above approaches don’t work when the integral limits a and b are not finite, i.e. a = -∞ and/or b = ∞. One approach for dealing with such cases is to use a transformation. For example, to deal with

when the lower limit a is finite but the upper limit is not, is to use the transformation

Transformation

Thus

Derivative of the transformation

Also, if u = 0 then x = ∞. If u = 1 then x = a. This means that

Transformation of the integral

When the lower limit is -∞, we can use the following transformation

Transformation for -infinity

Thus

Integral transformation for -infinity

When both limits are non-finite, we see that for any a

Sum of integrals

Two transformations

Simplifications

If we choose a = 0, we get

Integral transformation result

Real Statistics Function

The various estimation techniques described above have been implemented in the Real Statistics Resource Pack via the INTEGRAL function. Click here for more information about this function as well as numerous examples of how to use it in Excel.

Double Integration

Click here for information about how to perform double integration.

References

Wikipedia (2021) Numerical integration
https://en.wikipedia.org/wiki/Numerical_integration

Bourne, M. (2021) Simpson’s Rule. Interactive Mathematics
https://www.intmath.com/integration/6-simpsons-rule.php

Wikipedia (2021) Riemann sum
https://en.wikipedia.org/wiki/Riemann_sum

Leave a Comment