ELi5: If the “rate of change” of a function is a tangible way to understand derivatives, what is a similar way to understand integrals?

Question

I know it’s the “area under the curve”, but what does that mean exactly? Is there a physical or tangible way to explain it?

I understand that a derivative is rate of change at a specific point, and something like acceleration is rate of change of speed. But how can I visualize that speed is the “integral” of acceleration? What does that mean, and how does it relate to the area underneath?

In: 172

Answer 1

Here’s one for you: I had to estimate the amount of gravel to fill a ditch. The ditch varied in depth from one end to the other (increasing, though not perfectly regularly). I measured the depth of the ditch at regular intervals and interpolated a function from those points. The integral of that function was the amount of gravel I would need.