TweetFollow Us on Twitter

Simpsons Rule
Volume Number:9
Issue Number:10
Column Tag:Pascal workshop

Simpson’s Rule

An ingenious method for approximating integrals

By Marek Hajek, Incline Village, Nevada

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

About the author

Marek Hajek has been programming the Macintosh since 1989. He programmed two and a half years for Sierra Software Innovations where he wrote several in-house MacApp applications, participated in the development of SuperTEView, and the relational database engine - Inside Out II. Currently, he is receiving his bachelor's degree in Computer Science at the University of Nevada, Reno. He supports his college education by making useful programming tools - sorting/searching algorithms, and custom development. He welcomes your comments on this article either by phone at (702) 673-3341 or write to P.O. Box 7542, Incline Village, NV 89450.

Simpson's Rule, named after the great English mathematician Thomas Simpson, is an ingenious method for approximating integrals. If you don't know what integrals are used for, don't feel bad. Many college students who complete three semesters of calculus may be able to “compute” an integral, but won't know its practical application either. Computation of integrals is difficult to learn and easy to forget. [Many years out of school, I can attest to this! - Ed.]

Integrals are essential to the modern world. Practical applications of the integral are found in business, hydrostatics, highway construction, travel to the moon, solving of differential equations, and other branches of science. The first computer ever built was constructed to speed up ballistic missile trajectory calculations which meant solving a lot of integrals. [Given the forces imposed on a missile (gravity, thrust, wind resistance, etc.), integration is necessary to determine its path. - Tech. Ed.]

To illustrate the use of integrals, look at the curve in Figure 1.1a. The curve is described by the equation (1+X4). I want to compute the area of the shaded region. Notice, the area is between the curve, the x-axis, and the x-coordinates [-1,1]. The integral that will compute the area of the shaded region is in figure 1.1b. Anybody familiar with integrals will tell you that there is no known way to solve the integral abstractly (the quick and easy way).

Figure 1.1a

Figure 1.1b

If an integral can be solved on an abstract level, the computation is relatively easy. In practical applications, however, an integral can seldom be solved abstractly. That's where the Simpson's Rule finds its use. Thomas Simpson invented an equation, today called Simpson's Rule, which can be used to approximate an integral.

APPROXIMATION

The following line shows this equation in abstract form.

Looks complicated? First, take a look at figure 1.2.

Figure 1.2

In the approximation equation, the variables a and b are the boundaries of the integral and correspond to -1 and 1 in figure 1.1a. The variable n is the number of times the region under the curve is partitioned into smaller regions. You only have to know two things about n. First, the larger n is, the more accurate the approximation. And second, n must be a positive even integer (+2, 4, 6, ...). The function f(x) is the function you are integrating. In my example, it is (1+X4). Whenever you encounter f(x) in the equation, pass it the appropriate parameter. The parameters are the x-coordinates of the partitions (X0, X1, X2, X3, , Xn).

[Simpson’s rule approximates the function on each subinterval of the partition by a parabola that passes through the endpoints and the midpoint. The area under a parabola is easily calculated. Adding up these areas gives an estimate of the integral. - Tech. Ed.]

EXAMPLE COMPUTATION

To compute the integral in figure 1.1b, given four partitions (n=4), the approximation looks like this:

Simplified:

Simplified:

Result: 2.1791. . .

SIMPSON'S RULE - PASCAL

Figure 1.3

I translated Simpson's Rule into several pascal functions. To help you see what each function does, the equation is divided into three parts - Head, Twos/Fours, and First/Last (Figure 1.3). Have fun!

CODE LISTING

{--------------------Main Program----------------------------}
PROGRAM Simpson;
(* Author  - Marek Hajek *)
(* P.O. Box 7542 *)
(* Incline Village, NV 89450 *)

(* This program was written with Think Pascal 4.0.1 *)
 USES
(* Make sure you include the sane library *)
  Auxiliary, Sane;

 CONST
  kLowerLimit = -1;               (* Corresponds to "a" *)
  kUpperLimit = 1;                (* Corresponds to "b" *)
  kPartitions = 4;                (* Corresponds to n = 4 *)

 VAR
  result: Extended;
  (* The approximated result of the Integral *)

BEGIN
 ShowText;  (* Brings up the Think Pascal text window *)

 result := ComputeIntegral(kLowerLimit, kUpperLimit, 
   kPartitions, IntegrandFunction);
 writeln('Integral with lower/upper limits ', kLowerLimit : 0, 
   '/', kUpperLimit : 0, ', subintervals ', kPartitions : 0, 
   ' is: ', result);

 readln; (* Stop here before the text window disappears *)
END.

{--------------------ComputeIntegral-------------------------}
FUNCTION ComputeIntegral (lowerLimit, upperLimit: Extended;
       partitionCount: LongInt;
       FUNCTION IntegrandFunction (partitionCoordinate: 
       Extended): Extended): Extended;
(* The function ComputeIntegral calls the necessary *)
(* functions to compute the individual parts.*)
(* It returns the approximate result. *)
VAR
   result: Extended;
   head: Extended;
   partitionIncrement: Extended;
   partitionCoordinate: Extended;
   index: LongInt;

 BEGIN
  head := ComputeHead(lowerLimit, upperLimit, partitionCount);
  result := FirstAndLast(lowerLimit, upperLimit, 
    IntegrandFunction);

  partitionIncrement := 
    (upperLimit - lowerLimit) /  partitionCount;
  partitionCoordinate := lowerLimit;

(* The FOR  loop computes the second part of the *)
(* integral -> Twos/Fours *)
  FOR  index := 1 TO partitionCount - 1  DO
   BEGIN
(* Partition coordinate corresponds to X0, X1, X2,.....Xn *)
    partitionCoordinate := 
      partitionCoordinate +  partitionIncrement;

(* Odd index means compute 4* f(x), even index *)
(* means compute 2 * f(x)  *)
    IF Odd(index) THEN
     result := result + 
       4 * IntegrandFunction(partitionCoordinate)
    ELSE
     result := result + 
       2 * IntegrandFunction(partitionCoordinate)

   END;  (* FOR ... *)

  ComputeIntegral := head * result;
 END;

{------------------IntegrandFunction-------------------------}
 FUNCTION IntegrandFunction (partitionCoordinate: 
   Extended): Extended;
(* The Integrand function is the function inside the *)
(* integral and needs to be defined by you. In my example, *)
(* the integrand function is  (1+X4) and is translated *)
(* into pascal. The function takes one argument which is *)
(* the x coordinate of the partition. *)

 BEGIN
{ This functions computes ->  (X * X * X * X +1)  }
  IntegrandFunction := 
    SQRT(XpwrI(partitionCoordinate, 4) + 1);
 END;

{---------------------ComputeHead----------------------------}
 FUNCTION ComputeHead (lowerLimit, upperLimit: Extended;
       partitionCount: LongInt): Extended;
(* Computes the first part of the integral equation, *)
(* the Head.  Corresponds to (b - a)/(3*n)  *)

 BEGIN
  ComputeHead := 
    (upperLimit - lowerLimit) / (3 * partitionCount);
 END;

{----------------------FirstAndLast--------------------------}
 FUNCTION FirstAndLast (lowerLimit, upperLimit: Extended;
       FUNCTION IntegrandFunction (partitionCoordinate: 
       Extended): Extended): Extended;
(* Computes the third part of the integral, the *)
(* FIRST/LAST.  Corresponds to [f(X0) + f(Xn)  *)

 BEGIN
  FirstAndLast := IntegrandFunction(lowerLimit) + 
    IntegrandFunction(upperLimit);
 END;
 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Lyn 1.5.11 - Lightweight image browser a...
Lyn is a lightweight and fast image browser and viewer designed for photographers, graphic artists and Web designers. Featuring an extremely versatile and aesthetically pleasing interface, it... Read more
NeoOffice 2014.11 - Mac-tailored, OpenOf...
NeoOffice is a complete office suite for OS X. With NeoOffice, users can view, edit, and save OpenOffice documents, PDF files, and most Microsoft Word, Excel, and PowerPoint documents. NeoOffice 3.x... Read more
LaunchBar 6.4 - Powerful file/URL/email...
LaunchBar is an award-winning productivity utility that offers an amazingly intuitive and efficient way to search and access any kind of information stored on your computer or on the Web. It provides... Read more
Remotix 3.1.4 - Access all your computer...
Remotix is a fast and powerful application to easily access multiple Macs (and PCs) from your own Mac. Features Complete Apple Screen Sharing support - including Mac OS X login, clipboard... Read more
DesktopLyrics 2.6.6 - Displays current i...
DesktopLyrics is an application that displays the lyrics of the song currently playing in "iTunes" right on your desktop. The lyrics for the song have to be set in iTunes; DesktopLyrics does nothing... Read more
VOX 2.5.1 - Music player that supports m...
VOX is a beautiful music player that supports many filetypes. The beauty is in its simplicity, yet behind the minimal exterior lies a powerful music player with a ton of features and support for all... Read more
NetNewsWire 4.0.0 - RSS and Atom news re...
NetNewsWire is the best way to keep up with the sites and authors you read most regularly. Let NetNewsWire pull down the latest articles, and read them in a distraction-free and Mac-like way. Native... Read more
MacUpdate Desktop 6.0.6 - Search and ins...
MacUpdate Desktop 6 brings seamless 1-click installs and version updates to your Mac. With a free MacUpdate account and MacUpdate Desktop 6, Mac users can now install almost any Mac app on macupdate.... Read more
ForkLift 2.6.5 - Powerful file manager:...
ForkLift is a powerful file manager and ferociously fast FTP client clothed in a clean and versatile UI that offers the combination of absolute simplicity and raw power expected from a well-executed... Read more
Drive Genius 4.1.0 - Powerful system uti...
Drive Genius 4 gives you faster performance from your Mac while also protecting it. The award-winning and improved DrivePulse feature alerts you to hard drive issues before they become major problems... Read more

Biz Builder Delux (Games)
Biz Builder Delux 1.0.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0.0 (iTunes) Description: Ah, there's nothing like the rhythmic bustle of a burgeoning business burg... especially when you're the one building it... | Read more »
Auroch Digital is Bringing Back Games Wo...
| Read more »
Carbo - Handwriting in the Digital Age...
Carbo - Handwriting in the Digital Age 1.0 Device: iOS Universal Category: Productivity Price: $3.99, Version: 1.0 (iTunes) Description: | Read more »
Draggy Dead (Games)
Draggy Dead 1.1 Device: iOS Universal Category: Games Price: $.99, Version: 1.1 (iTunes) Description: Ditch your dead end job and take up a rewarding career in Grave Robbing today!Guide the recently deceased to a fun filled life of... | Read more »
Bad Dinos (Games)
Bad Dinos 1.0.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0.0 (iTunes) Description: | Read more »
The Apple Watch isn't Great as a Fi...
| Read more »
Show the World What You See With Stre.am...
Live broadcasting is getting popular on mobile devices, which is why you can now get Stre.am, by Infinite Takes. [Read more] | Read more »
PhotoTime's 2.1 Update Adds Apple W...
The latest PhotoTime update is adding even more functionality to the handy photo organizing app. Yep, including Apple Watch support. [Read more] | Read more »
Oh My Glob! Adventure Time Puzzle Quest...
Finn and Jake are taking over D3 Go!'s popular puzzle game series in the upcoming Adventure Time Puzzle Quest. [Read more] | Read more »
Earthcore: Shattered Elements - Tips, Tr...
At first glance, Earthcore: Shattered Elements seems like a rather simple card-battling game. Once you’re introduced to skills that will change quite a bit. Even more so once you start to acquire hero cards. But it’s not so complicated that we... | Read more »

Price Scanner via MacPrices.net

OtterBox Maximizes Portability, Productivity...
From the kitchen recipe book to the boarsroom presentation, the OtterBox Agility Tablet System turns tablets into one of the most versatile pieces of handheld technology available. Available now, the... Read more
Launch of New Car App Gallery and Open Develo...
Automatic, a company on a mission to bring the power of the Internet into every car, has announced the launch of the Automatic App Gallery, an app store for nearly every car or truck on the road... Read more
Memorial Day Weekend Sale: 13-inch 1.6GHz Mac...
Best Buy has the new 13″ 1.6GHz/128GB MacBook Air on sale for $849 on their online store this weekend. Choose free shipping or free local store pickup (if available). Sale price for online orders... Read more
Memorial Day Weekend Sale: 27-inch 3.5GHz 5K...
Best Buy has the 27″ 3.5GHz 5K iMac on sale for $2099.99 this weekend. Choose free shipping or free local store pickup (if available). Sale price for online orders only, in-store prices may vary.... Read more
Sale! 16GB iPad mini 3 for $349, save $50
B&H Photo has the 16GB iPad mini 3 WiFi on sale for $349 including free shipping plus NY sales tax only. Their price is $50 off MSRP, and it’s the lowest price available for this model. Read more
Price drop on 2014 15-inch Retina MacBook Pro...
B&H Photo has dropped prices on 2014 15″ Retina MacBook Pros by $200. Shipping is free, and B&H charges NY sales tax only: - 15″ 2.2GHz Retina MacBook Pro: $1799.99 save $200 - 15″ 2.5GHz... Read more
With a Mission to Make Mobile Free, Scratch W...
Scratch Wireless, claiming to be the world’s first truly free mobile service, has announced the availability of a new Scratch-enabled Android smartphone, the Coolpad Arise. The smartphone is equipped... Read more
First-Ever Titanium Alloy Curved iPhone 6 Scr...
One of the most common problems with mobile phones is damage to the screens. The slightest drop can cause a dreaded spider web of gashes and cracks in the glass panel surface that can cost $hundreds... Read more
Preorder new 12-inch MacBook, $10 off, save o...
Adorama has new 12″ Retina MacBooks available for preorder for $10 off MSRP including free shipping plus NY & NJ sales tax only. For a limited time, Adorama will include a free Apple USB-C to USB... Read more
Will iOS 9 Finally Bring Productivity Friendl...
Ah, the irony. From its original announcement in 2010, Apple has doggedly insisted that the iPad remain “simple,” thus arbitrarily limiting its considerable potential as a content creation and... Read more

Jobs Board

*Apple* Solutions Consultant - Retail Sales...
**Job Summary** As an Apple Solutions Consultant (ASC) you are the link between our customers and our products. Your role is to drive the Apple business in a retail Read more
Business Development Manager - *Apple* Pay...
**Job Summary** Apple Pay is seeking an experienced Business Development professional to join the Apple Pay team to develop partnerships and strategic alliances with Read more
Project Manager, *Apple* Retail New Store O...
**Job Summary** An Apple Retail New Store Openings & Remodels Project Manager is responsible for successfully managing the openings, remodels, and small works of Read more
SW QA Engineer - *Apple* TV - Apple (United...
**Job Summary** The Apple TV team is looking for experienced Quality Assurance Engineers with a passion for delivering first in class home entertainment solutions. **Key Read more
Partner Marketing Manager - *Apple* Pay - A...
**Job Summary** The Apple Pay partner marketing team is looking for a marketing manager to develop and drive US marketing programs with our financial institution Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.