These are useful in circuits in various applications. For example if you want to make certain waveform shapes, it can be convenient to integrate or differentiate- if you want a sawtooth, say, you could integrate a square wave; if you want a square wave from a sawtooth, you would differentiate.

Integration is useful if you have a circuit in which voltage is proportional to some quantity of interest, and you want to sum that quantity. An application from my own research: I detect photons using sensors for which voltage is proportional to amount of light detected; to measure the total amount of light, I want to integrate the signal.

Similarly, differentiating is useful if you want to determine the rate of change of voltage.

Kate Scholberg 2017-02-13