DISCRETE FUNCTIONS

INTERACTIVE E-BOOK

INTEGRATION

MAJOR

GENERAL

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Title	An interactive e-book for illustrating integration of discrete functions.
Creator	Autar K Kaw
Subject and Keywords	Discrete Functions, Numerical Integration, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	An interactive E-book for illustrating integration of discrete functions. It includes links to simulations in Mathcad, Maple, Mathematica and Matlab for the algorithm.
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/discrete_07int_ebook.pdf
Language	English
Rights	http://numericalmethods.eng.usf.edu/rights.htm

Example 1: Velocity as a function of time

Method 1: Average velocity method

Method 2: Trapezoidal Rule

Method 3: Polynomial interpolation

Method 4: Spline interpolation

Example 2: Finding Error

Method 1: Average velocity error

Method 2: Trapezoidal Rule error

Method 3: Polynomial interpolation error

Method 4: Spline interpolation error

Simulation

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

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 simplify the integral. Here, we will discuss the use of numerical integration with discrete data points, which can involve unequal segments.

Figure 1: Integration of a function

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INTEGRATING DISCRETE FUNCTIONS

The method of integrating discrete functions is shown below using an example.

Example 1

The upward velocity of a rocket is given as a function of time in Table 1.

Table 1: Velocity as a function of time

t	v(t)
s	m/s
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

Figure 1: Velocity vs. time data for the rocket example

Determine the distance, covered by the rocket from to using the velocity data provided and any applicable numerical technique.

Solution

Method 1: Average Velocity Method

The velocity of the rocket is not provided at and so we will have to use the interval that includes to find the average velocity of the rocket within that range. In this case, the interval will suffice.

Since we get

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Method 2: Trapezoidal Rule

If we were finding the distance traveled between times on the data table, we would simply find the area of the trapezoids with the corner points given as the velocity and time data points. For example

and applying the trapezoidal rule over each of the above integrals gives

The values of v(10), v(15) and v(20) are given in Table 1.

However, we are interested in finding

and applying the trapezoidal rule over each of the above integrals gives

How do we find v(11) and v(16)? We use linear interpolation. To find v(11),

and to find v(16)

Then

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Method 3: Polynomial Interpolation to find velocity profile

Because we are finding the area under the curve from we must use three points, and to fit a quadratic through the data. Using polynomial interpolation, our resulting velocity function is (refer to notes on direct method of interpolation)

Now, we simply take the integral of the quadratic within our limits, giving us

Method 4: Spline Interpolation to find the velocity profile

Fitting quadratic splines (refer to notes on spline method of interpolation) through the data obtains the following set of quadratics

The value of the integral would then simply be

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Example 2

What is the absolute relative true error for each of the above four methods if the above data in Table 1 was actually obtained from the velocity profile of

, where v is given in m/s and t in s.