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

It’s a way of continuously adding stuff.

Like if I give you a bucket of 1 gal of water, then a bucket of 2 gal, then a bucket of 3 gal, you can just do 1+2+3. But if I turn on a hose and the flow from hose is continuously increasing, then you need to add this up using an integral.