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 you have a curve. Now draw a big rectangle under it. The area of that rectangle is close to the area under the curve but not exactly. We can get closer by instead drawing 5 rectangles so that they are closer to the shape of the curve. If you have infinite rectangles then you will be able to add them up and get an accurate representation of the area that they fill. The math behind using an infinite number of rectangles has been simplified and named an integral.