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

Made me think of the “chemist integration” method: Plot the function, cut it out with scissors, measure on a scale 🙂

Seriously though, if you are OK with acceleration being the “rate of change” of speed, just reverse it: speed is the accumulation of acceleration over time.

It’s a simple multiplication for a constant (area of a rectangle), multiplication with the average for something linear (area of a rectangle plus a triangle), and for anything more complex, well, however the area under the curve grows over time.