Another way to look at this same idea is to say that the membrane potential integrates the current across it. Think of the membrane capacitance as a bucket, and the current flowing across the membrane as water filling the bucket. The height of the water level in the bucket is obviously proportional to how fast the water flows into it. This height is analogous to the membrane potential.
To be more realistic we should recognize that some current escapes in parallel from the capacitor, via the membrane resistance. Hence we can term our membrane a leaky integrator. This is like putting a hole in our bucket.