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

I’ve got an analogy that covers both derivatives and integrals!

Your function is the speed of your car at any given time. Plugging the time in the functions gives you the speed.

Derivative is your acceleration. Integral is the distance driven up until that point in time.