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

This is a really nice question which has highlighted a lack of clarity in my thinking. Having read through the comments for ultimate simplicity I am think I am I correct to say the integral is “what that achieved“.

so Integral of acceleration tells you the change in speed that particular acceleration achieved. Integral of speed tells you the distance covered (change in position that the speed achieved). Integral of the vol/second of water into the bucket is what that spray achieved (put x volume of water into the bucket). Integral of paycheck is what someone giving you money achieved over the period, ie your gross annual.