# 18. Anti-Derivative and Integral

## 18A. Introduction

Just as the operator d.1 can be applied to a function f to obtain its derivative, so the operator d._1 can be applied to obtain its anti-derivative, that is, a function whose derivative is f. For example:

```      derv=: d.1
anti=: d._1
```
```      c=: 1 2 3 4 5 6
c with p.
1 2 3 4 5 6&p.
c with p. derv
2 6 12 20 30&p.
c with p. anti
0 1 1 1 1 1 1&p.
c with p. anti derv
1 2 3 4 5 6&p.
```

Exercises

```   Experiment with the operators derv and anti on functions such as ^ and 1&o. and 2&o. and 6&o.
```
```   Compare the derivatives of the functions 1 2 3 4 5&p. and 7 2 3 4 5&p. , and explain their agreement.
```

## 18B. Area under a graph as a function

A study of the graph of Section 3B (an approximation to a circle) suggests that the area under the graph of a function f from the argument _1 (or any fixed point) to an argument x is itself a function of x (that depends upon the function f).

We will first make a similar graph, using a finer grid (with a spacing 0.05 between points):

```      a=:  i:1j40
PLOT a;cir a
```

What is the rate of change of the area function?

Consider the point x=: 0.5, the spacing s=: 0.05, and the next plotted point x+s. The change in area is due to the quadrilateral with base s and heights cir x and cir x+s, an area equal to s times the average of the heights cir x and cir x+s.

The rate of change ((cir x+s)-(cir x))%s is therefore the average of cir x and cir x+s. For small values of s, this average approaches cir x; the derivative of the area under the graph of cir is therefore the function cir itself.

In other words, the area under the graph of a function is the anti-derivative of the function. Since this area can be viewed as the aggregation or integration of the component areas, it is also called the integral of the function.

## 18C. Polynomial approximations

As illustrated in Section 3B, the area under a curve can be computed to give the value of the anti-derivative of a function at a chosen point. But this does not yield a function for the anti-derivative in the sense that the operator d._1 does for the functions to which it applies.

This situation is analogous to the equation-solving of Chapter 9, which gives the inverse of a function for some chosen point, but not the inverse function itself.

A practical solution to the anti-derivative of a function f is provided by finding the coefficients of a polynomial that approximates it, and then using the fact that the anti-derivative (as well as the derivative) of a polynomial is easily obtained.

The expression (f x) %. x^/i.n yields the coefficients of an n-term polynomial that best fits the function f at the points x. For example:

```      ]x=: i:1j10
_1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1
```
```      ]c=: (1&o. x) %. x ^/i.5
_3.46945e_17 0.997542 9.99201e_16 _0.156378 _7.77156e_16
```
```      c&p. _1 0 1
_0.841164 _3.46945e_17 0.841164
```
```      1&o. _1 0 1
_0.841471 0 0.841471
```
```      (c&p. x) ,. (sin x)
_0.841164 _0.841471
_0.717968 _0.717356
_0.564748 _0.564642
_0.389009 _0.389418
_0.198257 _0.198669
_3.46945e_17         0
0.198257  0.198669
0.389009  0.389418
0.564748  0.564642
0.717968  0.717356
0.841164  0.841471
```

The polynomial approximations may, however, be wildly wrong for arguments outside of the list x to which it was fitted.

The function der of Section 6B applied to a list of polynomial coefficients yields the coefficients of the derivative polynomial. A coresponding function for the anti-derivative may be defined as follows:

```      der=: 1: |. ] * i. on #
ant=: 0:,] % next i. on #
ant c
0 1 1 1 1 1 1
der ant c
1 2 3 4 5 6 0
```