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

If you plot the function, it’s the area between the line and the x axis. If you plot a straight line and find the integral, you will get the area of the triangle formed by the line, the x axis and a vertical line at the end. Look up a reimman sum, it might help you visualize it.

If that doesn’t help, it’s how much you’re accumulating. If your function is velocity, he integral is how far you’ve gone.