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Course: Calculus II
Topic: Integration Techniques
Subtopic: Numerical Integration

Overview

Recall Riemann Sums where we approximated the area under a curve by drawing rectangles and summing their area as the number of rectangles grew toward infinity. The act of using rectangles based on the left or right endpoint, or even based on the midpoint, caused some error in the area approximation. This error can be reduced by using a shape where the top of the rectangular strip more closely matches the curve such as a slanty line (which would make the rectangles into trapezoids) or a parabolic topper (used in Simpson's Rule). This is the idea behind the Trapezoid Rule and Simpson's Rule, both being methods of numerical integration where the definite integral is approximated using a numeric formula. These two formulas ("rules") are particular useful for approximating definite integrals that cannot be evaluated symbolically (e.g. their integrand's antiderivative is unknown).

Objectives

By the end of this topic you should know and be prepared to be tested on:

Terminology

Terms you should be able to define: partition, error, absolute error, relative error, error bound, and the two numerical integration rules below.

Numerical Integration Rules

Let `f(x)` be continuous on `[a,b]`. Partition `[a,b]` into `n` equal subintervals each having width `Delta x = (b-a)/n` such that `a = x_0 < x_1 < x_2 < ... < x_n = b` then `int_a^b f(x) dx` can be approximated by either of the following formulae.

Trapezoidal Rule

`T_n = (Delta x)/2 [f(x_o) + 2f(x_1) + 2f(x_2) + ... + 2f(x_(n-1)) + f(x_n)]`

Simpson's Rule

`S_n = (Delta x)/3 [f(x_o) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + ... + 4f(x_(n-1)) + f(x_n)]`


Numerical Integration Error Formulae

Suppose `c` is a computed numerical solution to a problem having an actual exact solution `x`. Two common measures of the error are absolute error `= |c-x|` and relative error `= |c-x|/|x|`, if `x!=0`.

Midpoint Rule Error Bound

Assume `f''` is continuous on `[a,b]` and `k` is a Real number such that `|f''(x)| <= k`, for all `x` on `[a,b]`. When approximating the integral `int_a^b f(x) dx` by the Midpoint Rule with `n` subintervals, the absolute error, `E_M` satisfies the inequality `E_M <= (k(b-a))/24 (Delta x)^2 = (k(b-a)^3)/(24n^2)`.

Trapezoidal Rule Error Bound

Assume `f''` is continuous on `[a,b]` and `k` is a Real number such that `|f''(x)| <= k`, for all `x` on `[a,b]`. When approximating the integral `int_a^b f(x) dx` by the Trapezoidal Rule with `n` subintervals, the absolute error, `E_T`, satisfies the inequality `E_T <= (k(b-a))/12 (Delta x)^2 = (k(b-a)^3)/(12n^2)`.

Simpson's Rule Error Bound

Assume `f^"(4)"` is continuous on `[a,b]` and `k` is a Real number such that `|f^"(IV)"(x)| <= k`, for all `x` on `[a,b]`. When approximating the integral `int_a^b f(x) dx` by Simpson's Rule with `n` subintervals, the absolute error, `E_S`, satisties the inequality `E_S <= (k(b-a))/180 (Deltax)^4 = (k(b-a)^5)/(180n^4)`.

Mini-Lectures and Examples

STUDY: Integration Techniques - Numerical Integration

Supplemental Resources (required!)

Evaluating numerical integration problems using these two rules can get quite messy. So while you need to know how to set-up and evaluate a numerical integration problem using the two rules, it is useful to check your work electronically using one of these two online calculators: Trapezoidal Rule Calculator and Simpson's Rule Calculator. They are particularly handy when the problem has a large n-value making the rules simply exhausting to solve symbolically. TIP: uncheck the "show steps" box so the decimal answer is readily available.

Supplemental Resources (recommended)

Krista King's Error Bounds for Midpoint, Trapezoidal, and Simpson's Rules is a down-to-earth readable explanation of the error formulae in action.

Supplemental Resources (optional)

Video: Numerical Integration, Selwyn Hollis's Video Calculus

Lesson: Approximating Definite Integrals, Dale Hoffman's Contemporary Calculus

Lesson: Approximating Definite Integrals, Paul Dawkins' Online Notes

rev. 2021-02-20