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

Integrals are adding up and combining all the (infinite) pieces into a whole.

It kinda matches up with the plain-english indea of integrating things, meaning to combine them all together.

* If you integrate a car’s speed (over time), you calculate how far it went, becuase you are adding up all the infinitesmal bits of motion it did at every infinitesmal moment in time.
* If you integrate a force (over time) you calculate how much it cause things to move (the ‘impulse’ or ‘change in momentum’), because you add up all the tiny microseconds of force being applied, together to get a total amount of movement you’ve caused
* If you integrate a circle (across a path) then you calculate the volume of a cylinder, because you add up the infinite number of circles that a cylinder is (or can be thought of as being) made up from

The ‘area under the graph’ is a convenient way to represent the number we are calculating in these cases, but those examples are the actual physical things the area represents.

[We can do integrals in more abstract cases too, but those are some actual physical instances.]