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

Imagine we have a rocket, and the rocket burns its fuel to go faster and faster and faster.

At any one time, it’s moving at a particular speed, has travelled a certain distance, and is accelerating a certain amount.

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These things have integral and derivative relationships with each other.

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Distance is the integral of speed, and speed is the derivative of distance.

Speed is the integral of acceleration, and acceleration is the derivative of speed.