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

Just flip your phrasing around. A derivative is the rate of change if a function. An integral of that derivative is just the current value of that function. Speed is the derivative of distance traveled. If you have a function of speed, the integral of the speed function is your total distance.