Definite Integrals

[Andreik]

I want to create a script and I need to compute defined integrals.

He's somebody who write a function to calculate defined integrals or how can I do that?

Thanks!

[Andreik]

I found an example but is written in C++

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/* Copyright (C) Henry Golding, 2008. 
 * All rights reserved worldwide.
 *
 * This software is provided "as is" without express or implied
 * warranties. You may freely copy and compile this source into
 * applications you distribute provided that the copyright text
 * below is included in the resulting source code, for example:
 * "Portions Copyright (C) Henry Golding, 2008"
 */
#include <cassert>

// Returns the area of a trapezoid of a given height where sideA 
// and sideB are the uneven sides
float CalculateTrapezoidArea(float sideA, float sideB, float height)
{
    // Area of a trapezoid: A = (a+b)h / 2 (Wikipedia)
    return ( ( (sideA+sideB) * height  ) / 2.0f );
}

// Approximates the area under the curve of function pointed to by pFunc
// (which must take and return a float) in the range x=start to x=end. 
// Accuracy is determined by number of steps used between these values. 
float ApproximateIntegral(float (*pFunc)(float), float start, float end, unsigned steps)
{
    // pFunc is a pointer to a function that takes and returns a float, which we will use as f(x).
    // The general method is to calculate points at step intervals and calculate area of
    // the trapezoid underneath then add areas together 

    int   i;    // counter
    float diff; // the difference between steps - used as trapezoid height later
    float* xValues = new float[steps + 2]; // start, end and steps in between
    float* yValues = new float[steps + 2]; // for calculated results

    // set start and end x values
    xValues[0] = start;
    xValues[steps + 1] = end;

    // Interpolate x values for number of steps
    // Loop from second element to penultimate element
    diff = (end - start) / (steps + 1);
    for (i = 1; i <= steps; ++i)        
    {
        xValues[i] = xValues[i - 1] + diff;
    }

    // now we have all the x values, calculate all corresponding y values (or f(x))
    for (i = 0; i < steps + 2; ++i)
    {
        yValues[i] = (*pFunc)(xValues[i]);
    }

    // now calculate the area under each trapezoid
    // a will be first y of pair, b will be second y, h will be diff from earlier
    float finalArea = 0.0f;

    // we always need to do n-1 traps where n is the number of points we have
    for (i = 0; i < steps + 1; ++i)
    {
        finalArea += CalculateTrapezoidArea(yValues[i], yValues[i+1], diff);
    }

    delete[] xValues;
    delete[] yValues;

    return finalArea;
}

// A test function which returns y = x^2 - 4
float test(float x)
{
    return (x * x) - 4.0f;
}

// Test harness for the integral calculation functions
int main()
{
    float result;

    // check trapezoid area
    // a = 5, b = 10, h = 3, result should = 22.5
    result = CalculateTrapezoidArea(5.0f, 10.0f, 3.0f);
    assert( result == 22.5f);

    // check integral calculation
    // result of y = x^2 - 4 is approx -10.67f between -2 and 2
    result = ApproximateIntegral(test, -2.0f, 2.0f, 1);
    assert( result == -8.0f ); // correct calculation but only one step so very inaccurate

    result = ApproximateIntegral(test, -2.0f, 2.0f, 1000);
    assert( result == -10.666637f ); //more accurate

    return 0;
}

I wrote this in AutoIt, but is not so accurate:

Func TrapezoidArea($a,$b,$h)
    Return ((($a+$b)*$h)/2)
EndFunc

Func Integral($func,$x1,$x2,$step)
    Local $AREA = 0
    $func = StringLower($func)
    For $INDEX = $x1 To $x2 Step $step
        $valA = Execute(StringReplace($func,"x","$INDEX"))
        $valB = $valA + $step
        $AREA += TrapezoidArea($valA,$valB,$step)
    Next
    Return $AREA
EndFunc

Edited November 10, 2008 by Andreik

[martin]

Maybe the inacuracy is because you are calculating from $x1 To $x2 but you should calculate from $x1 To $x2 - $Step.

[Andreik]

Now it's more better:

Func TrapezoidArea($a,$b,$h)
    Return ((($a+$b)*$h)/2)
EndFunc

Func Integral($FUNCTION,$X1,$X2,$STEP)
    Local $Area = 0
    Local $xValues[$STEP+2]
    Local $yValues[$STEP+2]
    $FUNCTION = StringLower($FUNCTION)
    $xValues[0] = $X1
    $xValues[$STEP+1] = $X2
    Local $DIFF = ($X2-$X1)/($STEP+1)
    For $INDEX = 1 To $STEP
        $xValues[$INDEX] = $xValues[$INDEX - 1] + $DIFF
    Next
    For $INDEX = 0 To $STEP+2
        If $INDEX >= $STEP+2 Then ExitLoop
        $yValues[$INDEX] = Execute(StringReplace($FUNCTION,"x","$xValues[$INDEX]"))
    Next
    For $INDEX = 0 To $STEP+1
        If $INDEX >= $STEP+1 Then ExitLoop
        $Area += TrapezoidArea($yValues[$INDEX],$yValues[$INDEX+1],$DIFF)
    Next
    Return Round($Area,5)
EndFunc

MsgBox(0,"",Integral("x",0,1,1000))

Edited November 10, 2008 by Andreik

[martin]

Your first approach looked simpler to me, but it needed a small change

Func TrapezoidArea($a,$b,$h)
    Return ((($a+$b)*$h)/2)
EndFunc

Func Integral($func,$x1,$x2,$step)
    Local $AREA = 0
    $func = StringLower($func)
    $last = Execute(StringReplace($func,"x",$x1))
    For $INDEX = $x1 + $step To $x2+$step Step $step
        $valA = Execute(StringReplace($func,"x","$INDEX"))
        $AREA += TrapezoidArea($valA,$last,$step)
        $last = $valA
    Next
     Return Round($Area,5)
EndFunc

MsgBox(0,"",Integral("x^2",0,1,.001000))

[Andreik]

Your first approach looked simpler to me, but it needed a small change

Func TrapezoidArea($a,$b,$h)
    Return ((($a+$b)*$h)/2)
EndFunc

Func Integral($func,$x1,$x2,$step)
    Local $AREA = 0
    $func = StringLower($func)
    $last = Execute(StringReplace($func,"x",$x1))
    For $INDEX = $x1 + $step To $x2+$step Step $step
        $valA = Execute(StringReplace($func,"x","$INDEX"))
        $AREA += TrapezoidArea($valA,$last,$step)
        $last = $valA
    Next
     Return Round($Area,5)
EndFunc

MsgBox(0,"",Integral("x^2",0,1,.001000))

It`s more simple then second example but the second example it`s still more accurate.

Maybe if I change Round($Area,5) with Round($Area,2) will be better.

Please test the second example , looks to run in real time with a good result.

BTW I like your changes, it`s more simple.

Edited November 10, 2008 by Andreik

[martin]

It`s more simple then second example but the second example it`s still more accurate.

Maybe if I change Round($Area,5) with Round($Area,2) will be better.

Please test the second example , looks to run in real time with a good result.

BTW I like your changes, it`s more simple.

The version I gave had

For $INDEX = $x1 + $step To $x2 + $step Step $step

It should be

For $INDEX = $x1 + $step To $x2  Step $step

But for some values of $step the final loop seems not to get executed, so to ensure it is, and that another calculation isn't made beyond $x2 I tried this

For $INDEX = $x1 + $step To $x2 + $step/2 Step $step

That seems to improve it. Your second method still gives slightly different results (perhaps more accurate) sometimes but I don't see why.

[Andreik]

The version I gave had

For $INDEX = $x1 + $step To $x2 + $step Step $step

It should be

For $INDEX = $x1 + $step To $x2  Step $step

But for some values of $step the final loop seems not to get executed, so to ensure it is, and that another calculation isn't made beyond $x2 I tried this

For $INDEX = $x1 + $step To $x2 + $step/2 Step $step

That seems to improve it. Your second method still gives slightly different results (perhaps more accurate) sometimes but I don't see why.

You are right, in my first example the distance between $x2-$step and $x2 is not executed, from here is the difference result.

In second example the range between $x1 and $x2 is divided exactly in several intervals, resulting greater accuracy.

Edited November 10, 2008 by Andreik

[Malkey]

This appears to work, also.

; Approximates the area under the curve of  $Equation
Local $Equation = "x^2 - 4"

Local $dfx = ApproxAreaUnderCurve(-2, 2, $Equation)
MsgBox(0, "", "Approximate Integral = " & $dfx)

; $Start,$End - The limits (start and end ) along x-axis
; $sEq is equation eg. "3*x^2 - 4*x + 4"
; $NoOfSteps ( Optional,default = 1000) Number of Trapezoids to use in calculation.
; (The higher the number of Trapezoids = The thinner the Trapezoids = The better the approximation of area)
; Returns the approximate area under a curve scribed by the equation, $sEq.
Func ApproxAreaUnderCurve($Start, $End, $sEq, $NoOfSteps = 10000)
    Local $xStep = ($End - $Start) / $NoOfSteps ; Individual step length along X-axis (Width of Trapezoid)
    ;ConsoleWrite(" $xStep = " & $xStep & @CRLF)
    Local $area
    Local $yLST = StringReplace($sEq, "x", "$x") ; Equation will return Left Side of Trapezoid (y value)
    ;ConsoleWrite( " $yLST = " & $yLST & @CRLF)
    Local $yRST = StringReplace($sEq, "x", "($x + $xStep)"); Equation will return Right Side of Trapezoid (y value)
    ;ConsoleWrite( " $yRST = " & $yRST & @CRLF)
    For $x = $Start To $End - $xStep Step $xStep
        $area += $xStep * (Execute($yLST) + Execute($yRST)) / 2 ; Accumulative areas of Trapezoids
    Next
    Return Round($area, 8)
EndFunc   ;==>ApproxAreaUnderCurve

[Andreik]

This appears to work, also.

; Approximates the area under the curve of  $Equation
Local $Equation = "x^2 - 4"

Local $dfx = ApproxAreaUnderCurve(-2, 2, $Equation)
MsgBox(0, "", "Approximate Integral = " & $dfx)

; $Start,$End - The limits (start and end ) along x-axis
; $sEq is equation eg. "3*x^2 - 4*x + 4"
; $NoOfSteps ( Optional,default = 1000) Number of Trapezoids to use in calculation.
; (The higher the number of Trapezoids = The thinner the Trapezoids = The better the approximation of area)
; Returns the approximate area under a curve scribed by the equation, $sEq.
Func ApproxAreaUnderCurve($Start, $End, $sEq, $NoOfSteps = 10000)
    Local $xStep = ($End - $Start) / $NoOfSteps ; Individual step length along X-axis (Width of Trapezoid)
    ;ConsoleWrite(" $xStep = " & $xStep & @CRLF)
    Local $area
    Local $yLST = StringReplace($sEq, "x", "$x") ; Equation will return Left Side of Trapezoid (y value)
    ;ConsoleWrite( " $yLST = " & $yLST & @CRLF)
    Local $yRST = StringReplace($sEq, "x", "($x + $xStep)"); Equation will return Right Side of Trapezoid (y value)
    ;ConsoleWrite( " $yRST = " & $yRST & @CRLF)
    For $x = $Start To $End - $xStep Step $xStep
        $area += $xStep * (Execute($yLST) + Execute($yRST)) / 2 ; Accumulative areas of Trapezoids
    Next
    Return Round($area, 8)
EndFunc   ;==>ApproxAreaUnderCurve

Just tested. Work fine. Thanks Malkey!