# Integral

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

TBD

## Notation

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

## Variants

There are several different algorithms that 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
6.28318530717958647692528676655900580189
○2x    ⍝ Exact answer
6.28318530717958647692528676655900576839
```

The Order of the Numerical Integration (the number of rectangles used to approximate the result) is, by default, 128 for the Gauss-Legendre algorithm, and 64 for Newton-Cotes. That number may be changed via the Variant operator as in

```      N←17x
⎕PP←60 ⋄ ⎕FPC←512
{÷1+⍵*2}∫⍠'g' N
1.51204050407917392632913838918797965663103904910367167450076
{÷1+⍵*2}∫⍠('g' 150) N
1.51204050407917392632913838918797965662193427820023884786854
{÷1+⍵*2}∫⍠('g' 170) N
1.51204050407917392632913838918797965662193428158710682038154
¯3○N    ⍝ Exact answer
1.51204050407917392632913838918797965662193428158710826434397
```

## 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 20 standard deviations on either side with more rectangles used in the approximation to yield a number within rounding error of the correct answer:

```      ¯20 nd∫⍠150 20
1.00000000000000000000000000000000000002
```

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.

## Numerical Differentiation

Note that Derivative, the inverse of this operator, has also been implemented.

## Acknowledgements

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