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
Think about what speed does to position. After a t amount of time with a constant velocity v, your position will have changed by v*t. If you picture them on a graph, where time is the horizontal axis and velocity is the vertical axis, there’s a horizontal line. The area under that line, a rectangle, is given by one side times the other, so v*t. As you can see, that area is equivalent to your change in position, or total distance travelled. It does not say anything about your current position, though, unless you know what it was originally, which is why the general solution for an integral always has that +c. In this case, c is your original position.
You can then also use this definition when the velocity isn’t constant and is instead some line whose integral, the area under it, represents the change in position for the given interval. You can extrapolate this to the other derivatives of position, too. The integral of acceleration will be the total change in speed.
Latest Answers