TRAPEZOIDAL RULE OF INTEGRATION AN INTERACTIVE EBOOK
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Title	An interactive e-book for illustrating the Trapezoidal Rule for numerical integration.
Creator	Autar K Kaw
Subject and Keywords	Trapezoidal Rule, Numerical Integration, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	This is an interactive E-book for illustrating Trapezoidal rule. It includes links to simulations in Mathcad, Maple, Mathematica and Matlab for the algorithm, multiple choice quizzes, problem set, and a PowerPoint presentation.
Publisher	Holistic Numerical Methods Institute, College of Engineering, University of South Florida, Tampa, FL 33620-5350.
Contributors	Autar Kaw, Michael Keteltas
Format	Text/HTML
Last Revised	June 14, 2004
Identifier	http://numericalmethods.eng.usf.edu/ebooks/trapezoidal_07int_ebook.pdf
Language	English
Rights	http://numericalmethods.eng.usf.edu/rights.htm

Trapezoidal Rule

Single-segment Trapezoidal Rule

Multiple-segment Trapezoidal Rule

Error in Multiple-segment Trapezoidal Rule

PowerPoint presentation

Derivations

Method 1: Derived from Calculus

Method 2: Also derived from Calculus

Method 3: Derived from Geometry

Method 4: Derived from Method of Coefficients

Method 5: Another approach on the Methods of Coefficients

Examples

Example 1: Single segment Trapezoidal Rule

Example 2: 2-segment Trapezoidal Rule

Example 3: Multiple-segment Trapezoidal Rule

Example 4: Multiple-segment Trapezoidal Rule

Simulations

Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Convergence [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Problem Sets

Multiple choice question examination

Problem Set: Discrete Functions

Problem Set: Continuous Functions

Background

This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website.

Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose.

Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here.

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What is integration?

Integration is the process of measuring the area under a function plotted on a graph. Why would we want to do so? Among the most common examples are finding the velocity of a body from acceleration functions, and displacement of a body from velocity data. Throughout the engineering fields, there are (what sometimes seems like) countless applications for integral calculus. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods have been developed to find the integral.

Here, we will discuss the trapezoidal rule of approximating integrals of the form

Figure 1: Integration of a function

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TRAPEZOIDAL RULE:

Trapezoidal rule is based on the Newton-Cotes formula that if one approximates the integrand by an n^th order polynomial, then the integral of the function is approximated by the integral of that n^th order polynomial. Integrating polynomials is simple and is based on the calculus formula.

(1)

So if we want to approximate the integral

(2)

to find the value of the above integral, one assumes

(3)

where

. (4)

where is an order polynomial. Trapezoidal rule assumes , that is, the area under the linear polynomial (straight line),

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DEREVATION OF THE TRAPEZOIDAL RULE:

Method 1: Derived from Calculus

Hence

. (5)

But what is a₀ and a₁? Now if one chooses, and as the two points to approximate by a straight line from to ,

(6)

(7)

Solving the above two equations for and ,

(8)

Hence from Equation (5),

(9)

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Method 2: Also derived from Calculus

can also be approximated by using the Newton’s divided difference polynomial as

(10)

Hence

(11)

This gives the same result as Equation (10) because they are just different forms of writing the same polynomial.

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Method 3: Derived from Geometry

The Trapezoidal rule can also be derived from geometry. Look at Figure 2. The area under the curve f₁(x) is a trapezoid. The integral

(12)

Figure 2: Geometric Representation

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Method 4: Derived from Method of Coefficients

Trapezoidal rule can also be derived by the method of coefficients. The formula

(13)

where

Figure 3: Area by Coefficients

The interpretation is that is evaluated at points and , and each function evaluation is given a weight of . Geometrically, Equation (12) is looked at as an area of a trapezoid, while Equation (13) is viewed as the sum of the area of two rectangles, as shown in Figure 3. How can one derive trapezoidal rule by the method of coefficients?

Assume

(14)

Let the right hand side be an exact expression for integrals of and , that is, the formula will then also be exact for linear combinations of and , that is, .

(15)

(16)

Solving the above two equations gives

(17)

Hence

. (18)

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Method 5: Another approach on the Method of Coefficients

Trapezoidal rule can also be derived by the method of coefficients by another approach

Assume

(19)

Let the right hand side be exact for integrals of the form

(20)

But we want

(21)

to give the same result as Equation (20) for .

(22)

Hence from Equations (20) and (22),

Since and are arbitrary for a general straight line

(23)

Again, solving the above two equations (23) gives

(24)

Therefore

(25)

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Example 1

The vertical distance covered by a rocket from to seconds is given by

a) Use single segment Trapezoidal rule to find the distance covered.

b) Find the true error, E_t for part (a).

c) Find the absolute relative true error for part (a).

Solution

a) , where

b) The exact value of the above integral is

so the true error is

c) The absolute relative true error, , would then be

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Multiple-segment Trapezoidal Rule:

In Example 1, the true error using a single segment trapezoidal rule was large. We can divide the interval [8, 30] into [8, 19] and [19, 30] intervals and apply Trapezoidal rule over each segment.

Hence

The true error,

The true error now is reduced from 807 m to 205 m. Extending this procedure to dividing into equal segments and apply the Trapezoidal rule over each segment, the sum of the results obtained for each segment is the approximate value of the integral.

Divide into equal segments as shown in Figure 4. Then the width of each segment is

(26)

Figure 4: Multiple (n=4) Segment Trapezoidal Rule

The integral I can be

broken into h integrals as

(27)

Applying Trapezoidal rule Equation (27) on each segment gives

………………

(28)

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Simulations

Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Convergence [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Example 2

The vertical distance covered by a rocket from to seconds is given by

a) Use two-segment Trapezoidal rule to find the distance covered.

b) Find the true error, E_t for part (a).

c) Find the absolute relative true error for part (a).

Solution:

a) The solution using 2-segment Trapezoidal rule is

b) The exact value of the above integral is

so the true error is

c) The absolute relative true error, , would then be

Table 1: Values obtained using multiple-segment Trapezoidal rule for

n	Value	E_t
1	11868	-807	7.296	---
2	11266	-205	1.853	5.343
3	11153	-91.4	0.8265	1.019
4	11113	-51.5	0.4655	0.3594
5	11094	-33.0	0.2981	0.1669
6	11084	-22.9	0.2070	0.09082
7	11078	-16.8	0.1521	0.05482
8	11074	-12.9	0.1165	0.03560

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Example 3:

Use Multiple-segment Trapezoidal Rule to find the area under the curve

from to .

Solution

Using two segments, we get

So what is the true value of this integral?

Making the absolute relative true error

Why is the true value so far away from the approximate values? Just take a look at Figure 5. As you can see, the area under the “trapezoids” (yeah, they really look like triangles now) covers a small the area under the curve. As we add more segments, the approximated value quickly approaches the true value.

Figure 5: 2-Segment Approx. of an Exp. Function

Table 2: Values obtained using Multiple-segment Trapezoidal Rule for

n	Approximate Value
1	0.681	245.91	99.724%
2	50.535	196.05	79.505%
4	170.61	75.978	30.812%
8	227.04	19.546	7.927%
16	241.70	4.887	1.982%
32	245.37	1.222	0.495%
64	246.28	0.305	0.124%

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Example 4:

Use multiple-segment Trapezoidal Rule to find

Solution

We cannot use Trapezoidal Rule for this integral, as the value of the integrand at is infinite. However, it is known that a discontinuity in a curve will not change the area under it. We can assume any value for the function at . The algorithm to define the function so that we can use multiple-segment Trapezoidal Rule is given below.

Function f(x)

If x = 0 Then f = 0

If x <> 0 Then f = x ^ (-0.5)

End Function

Basically, we are just assigning the function a value of zero at . Everywhere else, the function is continuous. This means the true value of our integral will be just that—true. Let’s see what happens using multiple-segment Trapezoidal Rule:

Using two segments, we get