# Integration

The inverse of differentiation. Mathematical process used in calculus.

### Historical Notes

1. ~250BC Archimedes ideas are far ahead of his contemporaries and include applications of an early form of integration.
2. 1647 Cavalieri publishes "Exercitationes geometricae sex" which contains in print for the first time the integral from 0 to a of xn.
3. 1656 Wallis publishes Arithmetica infinitorum which uses interpolation methods to evaluate integrals.
4. 1675 Leibniz uses the modern notation for an integral for the first time.
5. 1690 Jacob Bernoulli uses the word "integral" for the first time to refer to the area under a curve.
6. 1722 Work unfinished by Cotes on his death is published as Harmonia mensurarum - integration of rational functions and contains a thorough treatment of the calculus applied to logarithmic and circular functions.
7. 1739 D′Alembert publishes Mémoire sur le calcul intégral.

### Noise and Vibration

In vibration analysis, integration will convert an acceleration signal into a velocity signal, or a velocity signal into a displacement signal.

Integration can be done in the time or frequency domain. For this reason an accelerometer is the transducer of choice because velocity and displacement can be so easily derived from its output.

An analog integrator is actually a low pass filter with 6 dB of attenuation per octave. This is true of an analog integrator only above its low cutoff. And since the low cutoff cannot be zero, analog integrators have low-frequency limits, usually either 1 or 10 Hz.