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
I think it’s easier to start with position/velocity.
Say you have a car driving straight down a road at a constant velocity, let’s say 10m/s.
The derivative of our position function tells us how fast our position is changing, aka our speed. And in this case where our speed is constant that would just be a straight line at 10.
If our velocity function is a straight line, the integral is the area under that line and would tell us the distance we’ve traveled at any point in time. And that’s pretty easy to visualize since it’s just a rectangle.
Our width is 10, and at 1 second our length is 1. So the area under our line at 1 second is 10. In other words, we’ve traveled 10 meters.
And then again, at 2 seconds the area under the line is 10×2=20 meters. At 3 seconds it’s 10×3=30, etc. And that’s just a position function: Pos = 10*t
And that should make some sense intuitively too as your functions/lines get more complicated. If you draw a line showing the rate something is changing, when that line goes down the area under that line increases slower. If that line goes up, the area increases faster.
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