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 integral of a function is the sum of the function over a certain range. Often the range is with respect to a discrete quantity like time or distance. Sometimes the range is more abstract, like it might be all of the real numbers.

Anyway, a conceptual shorthand for the Integral of a function is just that it’s a sum of results of that function with respect to a set of calls to that function.