If the High Pass Filter is a differentiator, what's the analogy for Low Pass Filter?
The corollary is an integrator for a LPF. If you have a resistor feeding into a capacitor, then you get your input signal from the beginning of time integrated up to now.

Here's the math behind what I describe.

Band pass would be a LPF + a HPF so potentially an integrator feeding into a differentiator.