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

It’s an accumulation.

The more acceleration you add, and the longer you add it, the more speed builds up. Similarly the more speed you add for more time, the more distance builds up.

The “area under the curve” is really missing the point. Yes, the operations will give the same result but it’s not what you’re working out.