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 take the integral of the velocity function, you’ll get the position function as a result.

v = v_0 + at

where v = velocity, v_0 = initial velocity, a = acceleration, t = time

x = x_0 + v_0t + ½at²

Where x = position and x_0 = initial position.

Take the derivative if the position function with t as the variable. You’ll get the velocity function.

It helps to keep track of the units. Velocity is distance per time, acceleration is distance per time squared, position is distance. The units in the position function are: assuming metric

m = m + m/s * s + m/s² * s²

Same with the velocity function

m/s = m/s + m/s² * s

I struggled with calculus until I took physics.