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

Accumulation.

For example if you have a faucet filling a bucket and your function is how open the faucet is, the integral is how much water accumulated in the bucket.

Similarly with speed, if your function is acceleration, the integral is how much speed has built up.

The area under the curve is an accumulation of a bunch of thin vertical rectangles, each of which has a height equal to the function at the corresponding point.