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

>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?

It’s just a mathematical property. You just kinda gotta accept that. That’s kind of like asking, why is a number multiplied by 2 always going to be even?

We have found a way to demonstrate how different equations can relate to one another (derivatives and integrals). One of the ways derivatives relate is a function of where the equation you’re deriving *from* falls on a graph.

Take speed and its derivative, distance. Plot speed on a graph. Doesn’t even matter how the line looks, just as long as it’s speed on the y-axis and time on the x-axis.

For any snapshot on that graph, look at the Y-axis value. You see that value is “6 m/s”. So for this snapshot of 1 second on the X-axis, the Y axis says 6 m/s. SO, in that one second, how *far* did the object go? **6 m**.

Because the Y axis is meters/second, and the derivative of meters/sec is meters, then that 6 can demonstrate both the speed AND the derivative of that speed. So if you add ALL the speeds together (area under the graph), then you’re just adding up all the snapshots like the one I just did above.

I’ll just expand this out.

0sec-1sec: 6 m/s

1sec-2sec: 7 m/s

2sec-3sec: 8 m/s

How far did you go? 21 meters. If you plotted that and took the area underneath that 3 second graph of speed, you’d get 21. (Ignoring that you can’t instantaneously change speed)