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

In your example of linear motion, the integral is analagous to the distance covered between the two points in time. If the original function describes speed (distance over time), then differentiating it measure the rate of change of the rate of change of distance over time – the acceleration. You are adding an extra degree of time.

By integrating, you’re removing a degree of time; if you do this with speed, you’re just left with the distance.