Integral

 Z←{L} f∫ R returns the definite Integral of the function f between the points L and R.
L is an optional numeric singleton which represents the lower bound of the definite Integral. If it is omitted, 0 is used.
R is a numeric singleton which represents the upper bound of the definite Integral.
f is an arbitrary monadic function.

TBD

Variants

There are several different algorithms which may be used for Numerical Integration, two of which are Gauss-Legendre and Newton-Cotes. The default algorithm is Gauss-Legendre, however the faster but less accurate Newton-Cotes algorithm may be selected via the Variant operator as in

```      ⎕FPC←128
{1+1○⍵}∫⍠'g' ○2x  ⍝ Gauss-Legendre
6.28318530717958647692528676655900576839
{1+1○⍵}∫⍠'n' ○2x  ⍝ Newton-Cotes
6.28318530717958647692528676655899537749
6.28318530717958647692528676655900576839
```

The Order of the Numerical Integration (the number of rectangles used to approximate the result) is, by default, 50. That number may be changed via the Variant operator as in

```      N←17x
{÷1+⍵*2}∫⍠'g' N
1.51204050407629183188001289665203330309
{÷1+⍵*2}∫⍠('g' 60) N
1.51204050407917392727026418786910512212
{÷1+⍵*2}∫⍠('g' 70) N
1.51204050407917392633179507126314397112
{÷1+⍵*2}∫⍠('g' 80) N
1.512040504079173926329142042030891168
{÷1+⍵*2}∫⍠('g' 90) N
1.51204050407917392632913839289394666393
1.51204050407917392632913838918797965662
```

Examples

For example,

The Integral of {⍵*2} is {(⍵*3)÷3}, and so the Integral of that function from 0 to 1 is ÷3:

```      ⎕FPC←128
{⍵*2}∫1
0.333333333333333333333333333333333333334
```

The Integral of the 1+Sine function from 0 to ○2 is ○2:

```      {1+1○⍵}∫○2x ⋄ ○2x
6.28318530717958647692528676655900576839
6.28318530717958647692528676655900576839
```

and the Integral of the Sine function from 0 to ○2 is (essentially) 0

```      {1○⍵}∫○2x
¯5.12499200004882449667902954237053816394E¯40
```

A Normal Distribution is defined as nd←{(*¯0.5×⍵*2)÷√○2x}. Integrating it over the entire width from ¯∞ to yields an answer of 1 (the area under the curve). However, this Integration code doesn't handle infinities as yet, so instead we integrate the function over 10 standard deviations on either side to yield a number very close to the correct answer:

```      ¯10 nd∫ 10
0.999999999999207065378727724161962509243
```

Integrating this same function for one, two, and three standard deviations on either side yields the 3-sigma rule of :

```      ⍪¯1 ¯2 ¯3 nd∫¨ 1 2 3
0.682689492137085897170465091264075844955    ⍝ 68%
0.954499736103641585599434725666933125056    ⍝ 95%
0.997300203936739810946696370464810045244    ⍝ 99.7%
```

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

Acknowledgements

This feature is entirely based on Laurent Fousse's Numerical Integration code written in MPFR as described in this paper.