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 cumulative amount

Eg if the graph shows the number of litres per minute coming from a hose over time, the integral is the total amount of water that has been sprayed

And as you say the differential is whether, at any given time, the flow rate is increasing or decreasing