the integrator
Replace the feedback resistor of an inverting amplifier with a capacitor and the output becomes the running total of the input over time, the area accumulated under the input curve. Feed it a steady voltage and the output ramps smoothly; the longer the voltage is applied, the further the output travels, like water filling a tank at a rate set by the input.
The mechanism: the input resistor R turns Vin into a current Vin over R that flows into the virtual ground and charges the feedback capacitor C. Since voltage on a capacitor is the accumulated charge, Vout equals minus (1 over RC) times the integral of Vin with respect to time. A constant input Vin produces a linear ramp with slope minus Vin over RC. For example, with R of 100 kilohm and C of 1 microfarad, RC is 0.1 second, so a 1 V input ramps the output at minus 10 V per second.
Integrators are the engine of ramp and waveform generators, analog computers, control loops, dual-slope ADCs, and charge measurement. The honest caveat is drift: a pure integrator has infinite gain at DC, so even a tiny input offset or bias current is integrated forever and eventually drives the output into a rail. Real integrators place a large resistor across the capacitor to bound the DC gain, trading perfect integration for stability.
Drive the integrator with a square wave. While the input is high the output ramps down; while it is low the output ramps up. The result is a triangle wave, the time-integral of the square.
Integrating a square wave produces a triangle wave, a classic waveform-shaping trick.
A pure integrator's DC gain is infinite, so offset and bias current ramp it into the rail. In practice you shunt the capacitor with a resistor to tame the DC behavior.