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

Integral it’s just a sum (from the definition itself). If you have a quantity that adds several times. The integral is just that addition.

For example, if you have the rate of change of something, for example speed (distance per time unit), how would you find your position after some time? Yes! Adding that velocity over that time. You run at 10 km per hour. Then after 10 hours that would be 10km sumed ten times (100 kms). That’s the integral there, just a sum.