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

Speed is the integral of acceleration and position is the integral of speed. If you think of a rate of change as “subtracting” something concrete (where you actually are on the path between points A and B) to arrive at something more abstract, then the integral is doing the opposite of that – putting the concrete back in.

The most pressing need to compute integrals, historically, has been artillery firing tables. That’s because, given a gun of specified size, a propellant charge of specified power, and an artillery shell of specified weight, it’s pretty easy to determine how fast the shell is going when it leaves the cannon.

But firing artillery at an enemy isn’t about how fast the shell is going, it’s about how far it goes before it hits the ground. Because you’re trying to hit a particular position where the enemy is. So the integral is the act of taking the abstract rate of change in position of the shell over time (the speed) and computing the concrete change in its position altogether (where it actually lands.) That this needed to be approximated (there aren’t analytical solutions for most useful integrals) at increasing levels of accuracy stimulated the invention of the first electronic computers.