What exactly is an ``integrator'' or a ``differentiator''? Why are they called that?

An ``integrator'' is a circuit which will give you output proportional to the integral of $\hat{V}_{\rm in} (t)$, i.e., $\hat{V}_{\rm out} (t) = \int \hat{V}_{\rm in} (t) dt$. A low-pass filter at high frequency integrates to some approximation, although also attenuates the signal (we'll actually be seeing later different ways to make integrators using active components, so that you don't lose voltage). Similarly, a ``differentiator'' gives you the derivative of the input, $\hat{V}_{\rm out} = \frac{d \hat{V}_{\rm in}}{dt}$, and a high-pass filter at low frequency does this job. (Why would you want to do this, you may ask? See below.)