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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Integration started to make sense to me when I started thinking about it as really fancy multiplication. For example, if you know how fast you are going and how long you have been going for, you can figure out how far you have gone. This is simple when you are going the same speed the entire time, just multiply your speed times whatever time period you care about, and boom, you have it. But, if the speed is changing, you can’t do this directly. Instead, chop up time into tiny pieces, where your speed doesn’t change much, multiply the speed then by the tiny slice of time, and add up all of the distances you get with each slice. This works for really anything that is a derivative, or something changing based on something else changing. Your speed is distance traveled by change in time. Filling a bucket with water is changing volume by change in time. You could even do something like the change in surface area over change in volume when filling a balloon. Whatever the rate, integration let’s you get the top part of that rate by multiplying it by the bottom part, when you can’t multiply directly.
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