View source for Integral

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      <td valign="top"><apll>Z←{L} <i>f</i><_sg/> R</apll></td>
      <td></td>
      <td></td>
      <td>returns the definite Integral of the function <apll><i>f</i></apll> between the points <apll>L</apll> and <apll>R</apll>.</td>
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  <td><apll>L</apll> is an optional Real numeric singleton which represents the lower bound of the definite Integral.  If it is omitted, <apll>0</apll> is used.</td>
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  <td><apll>R</apll> is a Real numeric singleton which represents the upper bound of the definite Integral.</td>
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  <td><apll><i>f</i></apll> is an arbitrary monadic function whose argument and result are both Real numeric singletons.</td>
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==Introduction==

According to [https://en.wikipedia.org/wiki/Numerical_integration Wikipedia], "In analysis, '''numerical integration''' comprises a broad family of algorithms for calculating the numerical value of a definite integral".

This implementation uses several different algorithms to achieve this as well as extends these algorithms to Infinite and Left and Right semi-infinite intervals.

==Notation==

The symbol chosen for this operator is the Integral Sign (<apl><_sg/></apl>), entered from the keyboard as Alt-’S’ or Alt-Shift-'s' (U+222B), and used in mathematics for Integration.

==Applications==

There are many, many applications of Numerical Integration.  Here are but a few taken from [https://www.whitman.edu Whitman College]'s Mathematics Department's Calculus Online course:

<ol>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.01.html Area between curves]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.02.html Distance, Velocity, Acceleration]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.03.html Volume]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.04.html Average value of a function]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.05.html Work]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.06.html Center of Mass]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.07.html Kinetic energy; improper integrals]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.08.html Probability]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.09.html Arc Length]</li>
  <li>[https://www.whitman.edu/mathematics/calculus_online/section09.10.html Surface Area]</li>
</ol>

and more taken from the website [https://www.intmath.com Interactive Mathematics]:

<ol>
  <li>[https://www.intmath.com/applications-integration/1-apps-indefinite-integral.php Applications of the Indefinite Integral]</li>
  <li>[https://www.intmath.com/applications-integration/2-area-under-curve.php Area Under a Curve by Integration]</li>
  <li>[https://www.intmath.com/applications-integration/3-area-between-curves.php Area Between 2 Curves using Integration]</li>
  <li>[https://www.intmath.com/applications-integration/4-volume-solid-revolution.php Volume of Solid of Revolution by Integration]</li>
  <li>[https://www.intmath.com/applications-integration/shell-method-volume-solid-revolution.php Shell Method: Volume of Solid of Revolution]</li>
  <li>[https://www.intmath.com/applications-integration/5-centroid-area.php Centroid of an Area by Integration]</li>
  <li>[https://www.intmath.com/applications-integration/6-moments-inertia.php Moments of Inertia by Integration]</li>
  <li>[https://www.intmath.com/applications-integration/7-work-variable-force.php Work by a Variable Force using Integration]</a></li>
  <li>[https://www.intmath.com/applications-integration/8-electric-charges.php Electric Charges by Integration]</li>
  <li>[https://www.intmath.com/applications-integration/9-average-value-function.php Average Value of a Function by Integration]</li>
  <li>[https://www.intmath.com/applications-integration/10-force-liquid.php Force Due to Liquid Pressure by Integration]</a></li>
  <li>[https://www.intmath.com/applications-integration/11-arc-length-curve.php Arc Length of a Curve using Integration]</a></li>
  <li>[https://www.intmath.com/applications-integration/12-arc-length-curve-parametric-polar.php Arc Length of Curve: Parametric, Polar Coordinates]</a></li>
</ol>

==Variants==

There are several different algorithms that may be used for Numerical Integration; the main (and default) one is based upon the Tanh-Sinh Quadrature code in David H. Bailey's [https://www.davidhbailey.com/dhbsoftware/arprec-2.2.18.tar.gz ARPREC] package.  Other algorithms include [https://en.wikipedia.org/wiki/Gaussian_quadrature Gauss-Legendre] and [https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas Newton-Cotes] taken from Laurent Fousse's CRQ (Correctly Rounded Quadrature) [https://github.com/lfousse/crq Numerical Integration] code written in MPFR as described in [https://tel.archives-ouvertes.fr/tel-00477243/document this paper].  These algorithm may be selected via the Variant operator as in

<apll><pre>
      ⎕FPC←128
      {1+1○⍵}<_sg/>⍠'g' ○2<_x/>  ⍝ Gauss-Legendre
6.2831853071795864769252867665590057683<_mark>9</_mark> 
      {1+1○⍵}<_sg/>⍠'n' ○2<_x/>  ⍝ Newton-Cotes
6.283185307179586476925286766559005<_mark>7</_mark>1385
      ○2<_x/>    ⍝ Exact answer
6.28318530717958647692528676655900576839
</pre></apll>

The Order of the Numerical Integration (the number of rectangles used to approximate the result) is, by default, <apll>128</apll> for the Gauss-Legendre algorithm, and <apll>32</apll> for Newton-Cotes.  That number may be changed via the Variant operator as in

<apll><pre>
      N←17<_x/>
      ⎕PP←60 ⋄ ⎕FPC←512
      {÷1+⍵*2}<_sg/>⍠'g' N
1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
      {÷1+⍵*2}<_sg/>⍠'n' N
1.512040504079173926329138<_mark>3</_mark>4207540700696171248671126465236073
      {÷1+⍵*2}<_sg/>⍠('n' 64) N
1.51204050407917392632913838918797965662193428158<_mark>7</_mark>22152948792
      {÷1+⍵*2}<_sg/>⍠('n' 96) N
1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
      ¯3○N    ⍝ Exact answer
1.51204050407917392632913838918797965662193428158710826434397 
</pre></apll>

<p>There are additional choices for algorithms such as</p>

<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
<tr>
  <th align="left">Algorithm</th>
  <th align="left">Syntax</th>
  <th align="left">Source</th>
</tr>

<tr>
  <td>Gauss-Legendre</td>
  <td><apll><_sg/>⍠'G'</apll></td>
  <td>David H. Bailey's ARPREC ARPREC C++ code</td>
</tr>

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  <td>Error Function</td>
  <td><apll><_sg/>⍠'E'</apll></td>
  <td>David H. Bailey's ARPREC ARPREC C++ code</td>
</tr>

<tr>
  <td>Tanh-Sinh</td>
  <td><apll><_sg/>⍠'T'</apll></td>
  <td>David H. Bailey's ARPREC ARPREC C++ code</td>
</tr>

<tr>
  <td valign="top">Tanh-Sinh</td>
  <td valign="top"><apll><_sg/>⍠'t'</apll> and <apll><_sg/></apll></td>
  <td valign="top">David H. Bailey's ARPREC ARPREC C++ code<br/>translated to C &mdash; it is the default algorithm</td>
</tr>

<tr>
  <td>Gauss-Legendre</td>
  <td><apll><_sg/>⍠'g'</apll></td>
  <td>Laurent Fousse's CRQ code</td>
</tr>

<tr>
  <td>Newton-Cotes</td>
  <td><apll><_sg/>⍠'n'</apll></td>
  <td>Laurent Fousse's CRQ code</td>
</tr>

<tr>
  <td>Tanh-Sinh</td>
  <td><apll><_sg/>⍠'d'</apll></td>
  <td>Graeme Dennes Excel Spreadsheet code</td>
</tr>
</table>
<br/>

==Default Algorithm==

<p>When the Variant operator is applied to the default algorithm, the first number is used to replace the exponent <apll>N=8</apll> in the calculation of the outer loop limit: <apll>12×2*N</apll>.  Increasing this value (as in <apll><_sg/>⍠N</apll>) provides greater precision at the cost of performance.  Additionally, two levels of precision (and via another calculation, Comparison Tolerance) are used in this algorithm, one for certain internal values (e.g., abscissa and weights) and the other for more precise calculations (twice the first precision).  By default, the precision for less precise calculations is the current value of <apll>⎕FPC</apll>.  The Comparison Tolerance values are expressed as negative integers to a power of <apll>10</apll> as in <apll>10*-1+⌊⎕FPC×10⍟2</apll>.  It may be changed for any particular calculation by supplying a second numeric argument to the Variant operator as in <apll><_sg/>⍠(N1 N2)</apll>, where the first number is a positive integer used as the loop limit exponent and the second number is the negative integer power of <apll>10</apll>, i.e. it replaces <apll>-1+⌊⎕FPC×10⍟2</apll> in the above statement.</p>

<p>Not all of these additional algorithm's may be supported in future versions.</p>

==Examples==

The formula for the [https://en.wikipedia.org/wiki/Parabola#Similarity_to_the_unit_parabola Unit Parabola] is <apll>{⍵*2}</apll> whose Integral is <apll>{1<_r/>3×⍵*3}</apll>, and so the Integral of the Unit Parabola from <apll>-a</apll> to <apll>a</apll> is <apll>-/{1<_r/>3×⍵*3}a,-a ←→ 2<_r/>3×a*3</apll>:

<apll><pre>
      ⎕FPC←128
      a←2 ⋄ (-a){⍵*2}<_sg/>a
5.33333333333333333333333333333333333334
      2<_r/>3×a*3
16<_r/>3     ⍝ ←→ 5.33333333333333333333333333333333333333333333333333333333333
</pre></apll>

This may be used to calculate the [https://en.wikipedia.org/wiki/Parabola#Area_enclosed_between_a_parabola_and_a_chord Area Enclosed Between a Parabola and a Chord], the formula for which is two-thirds of the area of the surrounding rectangle of the chord, the line tangent to the Vertex, and the two sides <apll>x=-a</apll> and <apll>x=a</apll>.

Because the Unit Parabola curves upward and away from the X-axis, the above Integral returns the area under (i.e. below and outside) the Parabola.  In this case, because we're Integrating from <apll>-a</apll> to <apll>a</apll>, the chord is the line <apll>y={⍵*2} a ←→ y=a*2</apll>.  The top of the rectangle (chord) is of length <apll>a--a ←→ 2×a</apll> and of height <apll>y ←→ a*2</apll>, whose area is <apll>(2×a)×a*2 ←→ 2×a*3</apll>.  Subtracting the result of the Integral from the rectangle's area and multiplying this by two-thirds yields the area enclosed between a chord and a Parabola.

<apll><pre>
      2<_r/>3×(2×a*3)-(-a){⍵*2}<_sg/>a
7.1111111111111111111111111111111111111
</pre></apll>

The Integral of the 1+Sine function from <apll>0</apll> to <apll>○2</apll> is <apll>○2</apll>:

<apll><pre>
      {1+1○⍵}<_sg/>○2<_x/> ⋄ ○2<_x/>
6.283185307179586476925286766559005768<_mark>3</_mark>4
6.28318530717958647692528676655900576839
</pre></apll>

and the Integral of the Sine function from <apll>0</apll> to <apll>○2</apll> is (essentially) <apll>0</apll>

<apll><pre>
      {1○⍵}<_sg/>○2<_x/>
6.69453502284549420282962009846618284241<_E/>¯42
</pre></apll>

A [https://en.wikipedia.org/wiki/Normal_distribution Normal Distribution] is defined as <apll>nd←{(*¯0.5×⍵*2<_x/>)÷√○2<_x/>}</apll>.  Because this instance of the algorithm is scaled properly, integrating it over the entire width from <apll>¯∞</apll> to <apll>∞</apll> yields an answer of <apll>1</apll> (the area under the curve):

<apll><pre>
      ¯∞ nd<_sg/> ∞
1
</pre></apll>

Integrating this same function for one, two, and three standard deviations on either side yields the [https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule 3-sigma rule] of :

<apll><pre>
      ⍪¯1 ¯2 ¯3 nd<_sg/>¨ 1 2 3
0.682689492137085897170465091264075844958    ⍝ 68%
0.954499736103641585599434725666933125059    ⍝ 95%
0.997300203936739810946696370464810045244    ⍝ 99.7%
</pre></apll>

which describes about how many of the values in a normal distribution lie within one, two, and three standard deviations from the mean.

==Numerical Differentiation==

Note that [[Derivative]], the inverse of this operator, has also been implemented.  These two features may be combined in various ways, for example, to calculate [https://en.wikipedia.org/wiki/Arc_length#Finding_arc_lengths_by_integrating Arc Length].

The formula for Arc Length is simply <apll>arclen←{4○⍺⍺∂⍵}</apll> which can be used on any given function.  For example, the upper half of a unit circle is represented by the function <apll>{0○⍵}</apll>.  The interval <apll>[-√0.5, √0.5]</apll> is then a quarter of a circle, whose length is

<apll><pre>
      ⎕FPC←512 ⋄ ⎕PP←60
      (-√0.5<_x/>){0○⍵}arclen<_sg/>√0.5<_x/>
1.5707963267948966192313216916397514420985846996875529104874<_mark>7</_mark>
      ○0.5<_x/>     ⍝ Exact length = 1<_r/>2<_pi/>1
1.57079632679489661923132169163975144209858469968755291048747</pre></apll>

The arc length of the Unit Parabola from <apll>0</apll> to <apll>1</apll> is

<apll><pre>
      {⍵*2}{4○⍺⍺∂⍵}∫1
1.4789428575445974338279060194339144350716974305950082518812<_mark>1</_mark>
      (1<_r/>2×(√5<_r/>16)÷1<_r/>4)+1<_r/>4×⍟(1<_r/>2+√5<_r/>16)÷1<_r/>4       ⍝ Exact length (from Wikipedia, "Parabola#Arc_length")
1.47894285754459743382790601943391443507169743059500825188121
</pre></apll>