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

The rate of change *is* the derivative of a function outside of pathological cases. The area under the curve is the standard integral of the function. If you filled in with a crayon the full area under it, and tried to calculate that area, you’d end up with the integral or anti-derivative.