# INTEG

## Purpose:

Calculates the integral of a series using
Simpson’s rule.

## Syntax:

INTEG(series)

series |
- |
An interval or XY series, the series to integrate. |

## Returns:

A series or table.

## Example:

W1: gsin(1000,.001)

W2: integ(w1)

W3: (1 - cos(2*pi*xvals(w1)))/(2*pi)

W4: w2 - w3

W2 contains the integral of the 1 Hertz
sinewave in W1 over one period. The value of each point in W2 is the definite
integral of W1 up to the corresponding point in W1. W4 displays the difference
between the calculated integration and the analytical result in W3.

The value of this integral can be solved
analytically. In general:

For the specific case, we have:

For *t*_{0} = 0 and f
= 1:

This result is displayed in W3 and the
Simpson’s rule approximation is displayed in W2.

## Example:

y1 = gnorm(1000, .001);

a1 = integ(y1);

a2 = area(y1);

Series
a1 contains the cumulative area of y1
and scalar a2 contains the value
of the total area of y1. Note
that a1[end] == a2, the last point
of the cumulative area equals the total area.

## Example:

W1: gsin(1000,.001)

W2: xy(xvals(w1), yvals(w1))

W3: integ(w1)

W4: integ(w2)

W5: w3 - w4

W1 contains a 1 Hertz sinewave over one
period as an interval series. W2 converts W1 into an XY series. The X
values of W1 are implicit and can be constructed from the delta X value.
The X values of W2 are explicit.

The definite integrals of each series
are calculated and the differences are displayed in W5.

## Remarks:

INTEG uses the composite Simpson's rule
to compute the integral of evenly (interval series) or unevenly (XY series)
spaced data. This method fits a quadratic polynomial to three points of
the series and performs polynomial integration. In particular:

where

See CUMTRAPZ
for an implementation of the trapezoidal rule.

See AREA
to compute the total area.

See FINTEG
for an implementation in the frequency domain.

The input series can be Real or Complex.

## See Also:

AREA

CAREA

CUMTRAPZ

DERIV

FDERIV

FINTEG

LDERIV

PARTSUM

RDERIV

TRAPZ