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;
 
AAPL
$100.89
Apple Inc.
-0.65
MSFT
$45.01
Microsoft Corpora
-0.17
GOOG
$577.86
Google Inc.
-2.34

MacTech Search:
Community Search:

Software Updates via MacUpdate

BBEdit 10.5.12 - Powerful text and HTML...
BBEdit is the leading professional HTML and text editor for the Mac. Specifically crafted in response to the needs of Web authors and software developers, this award-winning product provides a... Read more
Microsoft Office 2011 14.4.4 - Popular p...
Microsoft Office 2011 helps you create professional documents and presentations. And since Office for Mac 2011 is compatible with Office for Windows, you can work on documents with virtually anyone... Read more
TextWrangler 4.5.10 - Free general purpo...
TextWrangler is the powerful general purpose text editor, and Unix and server administrator's tool. Oh, and also, like the best things in life, it's free. TextWrangler is the "little brother" to... Read more
BitTorrent Sync 1.4.72 - Sync files secu...
BitTorrent Sync allows you to sync unlimited files between your own devices, or share a folder with friends and family to automatically sync anything. File transfers are encrypted. Your information... Read more
Cyberduck 4.5.2 - FTP and SFTP browser....
Cyberduck is a robust FTP/FTP-TLS/SFTP browser for the Mac whose lack of visual clutter and cleverly intuitive features make it easy to use. Support for external editors and system technologies such... Read more
Tinderbox 6.0.3 - Store and organize you...
Tinderbox is a personal content management assistant. It stores your notes, ideas, and plans. It can help you organize and understand them. And Tinderbox helps you share ideas through Web journals... Read more
Adobe Photoshop CC 2014 15.1.0 - Profess...
Photoshop CC 2014 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Photoshop customer). Photoshop CS6 is still available for purchase (... Read more
Adobe InDesign CC 2014 10.0.0.70 - Profe...
InDesign CC 2014 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous InDesign customer). InDesign CS6 is still available for purchase (... Read more
Adobe InCopy CC 2014 10.0.0.70 - Create...
InCopy CC 2014 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous InCopy customer). InCopy CS6 is still available for purchase (without a... Read more
Adobe Illustrator CC 2014 18.0 - Profess...
Illustrator CC 2014 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Illustrator customer). Illustrator CS6 is still available for... Read more

Latest Forum Discussions

See All

Appointment With F.E.A.R. (Games)
Appointment With F.E.A.R. 1.02 Device: iOS Universal Category: Games Price: $2.99, Version: 1.02 (iTunes) Description: Travel back to the 1980s as a superpowered defender of justice in this all-new edition of Steve Jackson's... | Read more »
8bit Doves (Games)
8bit Doves 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: “I dove it” – Pocket Gamer “They told me I could be whatever I want, so I became a flying hobo” – Cactus Wolf “Let’s you crash into... | Read more »
Madden NFL Mobile Review
Madden NFL Mobile Review By Blake Grundman on August 26th, 2014 Our Rating: :: FINALLY GETS IT RIGHTUniversal App - Designed for iPhone and iPad It looks like EA Sports has finally discovered how to tread the line between free-to-... | Read more »
Superhero Workout Review
Superhero Workout Review By Jennifer Allen on August 26th, 2014 Our Rating: :: SUPERCHARGEDUniversal App - Designed for iPhone and iPad Want a workout with a kind of story tied into it? Superhero Workout is what you’re after.   | Read more »
Star Wars: Commander Review
Star Wars: Commander Review By Jennifer Allen on August 26th, 2014 Our Rating: :: CLASH OF THE FORCEUniversal App - Designed for iPhone and iPad It’s a lot like Clash of Clans, but the use of the Star Wars license means you don’t... | Read more »
Ridge Racer Slipstream Update Adds New C...
Ridge Racer Slipstream Update Adds New Cars, Challenges, and Tracks Posted by Ellis Spice on August 26th, 2014 [ permalink ] | Read more »
Infinity Blade III: Kingdom Come, the Fi...
Infinity Blade III: Kingdom Come, the Final Update, Set to Arrive September Posted by Ellis Spice on August 26th, 2014 [ permalink ] | Read more »
Furious Climber Review
Furious Climber Review By Jordan Minor on August 26th, 2014 Our Rating: :: REACH FOR THE STARSUniversal App - Designed for iPhone and iPad Furious Climber’s rough packaging can’t ruin its awesomely awkward gameplay   | Read more »
Don’t Forget the Milk – Let Heap, the Gr...
Don’t Forget the Milk – Let Heap, the Grocery List App, Help You Stay Organized While Shopping Posted by Jessica Fisher on August 26th, 2014 [ permalink ] | Read more »
MZR Review
MZR Review By Rob Thomas on August 26th, 2014 Our Rating: :: MAZED AND CONFUSEDUniversal App - Designed for iPhone and iPad How quickly can you escape from MZR’s neon mazes of doom? Why not find out for yourself?   | Read more »

Price Scanner via MacPrices.net

Life Inventory iOS Apps – Learn to Know Thyse...
James Hollender’s Life Inventory apps s are now on sale with 20% off thru Labor Day, 09/01/2014. This is a great opportunity to get started on that Moral Inventory you’ve been putting off doing for... Read more
Pocket Watch, LLC. Reveals Cloud Server For P...
Beaumont, Texas based Pocket Watch, LLC. has announced the availability of its new ActivePrint Cloud Server Powered by Raspberry Pi. With this small standalone box almost any USB printer or available... Read more
902it Simplifies Area Code Changes For Nova S...
The east coast Canadian provinces of Nova Scotia and Prince Edward Island are phasing in 10 digit telephone dialing, to be fully in place by November, in order to accommodate a second area code to... Read more
Boomerang iPad Stand Mounts Your iPad Anywher...
Boomerang, a Mountable Stand with Multiple Viewing Angles, is now available for iPad Air. Boomerang combines several functions that aim to expand your iPad’s potential in one, elegant product. The... Read more
Retina MacBook Pros available starting at $10...
The Apple Store has Apple Certified Refurbished 13″ and 15″ MacBook Pros available starting at $929. Apple’s one-year warranty is standard, and shipping is free: - 13″ 2.5GHz MacBook Pros (4GB RAM/... Read more
Apple 27-inch Thunderbolt Display (refurbishe...
The Apple Store has Apple Certified Refurbished 27″ Thunderbolt Displays available for $799 including free shipping. That’s $200 off the cost of new models. Read more
Apple offers free $25 iTunes gift card with p...
The Apple Store is offering a free $25 iTunes Gift Card with the purchase of a $99 Apple TV for a limited time. Shipping is free. Read more
Apple’s 2014 Back to School promotion: $100 g...
 Apple’s 2014 Back to School promotion includes a free $100 App Store Gift Card with the purchase of any new Mac (Mac mini excluded), or a $50 Gift Card with the purchase of an iPad or iPhone,... Read more
iPhone Camera vs. Top Tier DSLR In Picture Qu...
If you’ve wondered just how good the iPhone 5s’s 8 megapixel rear-facing camera is in terms of image quality, The Verge has posted an excellent feature that should remove any ambiguity on the matter... Read more
Apple Announces Limited iPhone 5 Defective Ba...
Apple says it has determined that a very small percentage of iPhone 5 devices may suddenly experience shorter battery life or need to be charged more frequently. The affected iPhone 5 units were sold... Read more

Jobs Board

Senior Event Manager, *Apple* Retail Market...
…This senior level position is responsible for leading and imagining the Apple Retail Team's global event strategy. Delivering an overarching brand story; in-store, Read more
*Apple* Solutions Consultant (ASC) - Apple (...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
Project Manager / Business Analyst, WW *Appl...
…a senior project manager / business analyst to work within our Worldwide Apple Fulfillment Operations and the Business Process Re-engineering team. This role will work Read more
Position Opening at *Apple* - Apple (United...
…customers purchase our products, you're the one who helps them get more out of their new Apple technology. Your day in the Apple Store is filled with a range of Read more
Position Opening at *Apple* - Apple (United...
**Job Summary** Being a Business Manager at an Apple Store means you're the catalyst for businesses to discover and leverage the power, ease, and flexibility of Apple Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.