Owl: A rooftop cistern fills from a rain gutter. The inflow rate r(t) (liters per hour) is recorded from noon onward. At 3 p.m. someone asks: how much water was added since noon?
That question is not "what is the height of the rate graph at 3?" — height is a rate. We need the accumulated contribution of that rate over three hours.
If r were constant, the answer would be rate times time — a rectangle. When r varies, we still accumulate, but in thin time slices.
Look at the whiteboard sketch. How would you estimate how much was added from noon to 3 if you only had a few rate readings?