In this subsection, we'll only look at integrating a single input signal. However, the circuit and analysis given here can be extended as above to create a circuit that will integrate a weighted sum. The integrator is shown in Figure 7.
Figure 7: An Integrator
Following the same type of analysis given above, the equating of input and feedback currents gives
Multiplying by and integrating both sides gives
Those in whose minds Calculus is fresh will notice that we've been a bit sloppy here. It is much more accurate to write
where c is an initial condition and is the point in time when the integrator goes into operation. We will usually take and always discuss time relative to that ``beginning of time.'' We'll have to be more careful if the computer causes itself to be reset with new initial conditions at various points during the computation. In those cases, we'll call the beginning of the whole computation time 0 and talk in relative terms about the intervals between resets.