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?
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The rate of change of a curve is going to be the angle that the line makes on that curve, indicating how much change is happening.
As that rate of change increases, as in its tilted up more than its tilted down, and you go farther along the X axis, the area under it increases, which is to say that its the product or just multiplying the overall effect of that rate of change with the progress along the x, plus wherever the curve started.
I’m not sure if that makes it more intuitive.
An ever increasing addition to the action that will reduce the error.
When your car continues to move towards the ditch, you add more and more corrective action (steer towards the center of the road).
I was first explained it as “the integral is the shadow” when you think of things dimensionally you’re rejoicing a dimension of it
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