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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It’s the inverse of derivative. If you use derivative to get speed from distance over time, you can use integral to get distance from speed. That’s why you need to add the constant, to tell how much you had moved before the integration period.
Think of it like the “work done”, for example, the integral of a velocity (m/s) is the distance traveled (m) up to that point (s)
The integral of a function is the sum of the function over a certain range. Often the range is with respect to a discrete quantity like time or distance. Sometimes the range is more abstract, like it might be all of the real numbers.
Anyway, a conceptual shorthand for the Integral of a function is just that it’s a sum of results of that function with respect to a set of calls to that function.
Integral it’s just a sum (from the definition itself). If you have a quantity that adds several times. The integral is just that addition.
For example, if you have the rate of change of something, for example speed (distance per time unit), how would you find your position after some time? Yes! Adding that velocity over that time. You run at 10 km per hour. Then after 10 hours that would be 10km sumed ten times (100 kms). That’s the integral there, just a sum.
Go backwards from speed which is a rate of change of location.
Using integrals we can determine the distance travelled between two defined points in time.
Integral is a sum of all the function values in given argument range. If you have a constant function y(x) = c, its integral for x from a to b will be c*(b – a). That’s exactly the area size below this function graph. Same aplies to other function shapes as well.
>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?
It’s just a mathematical property. You just kinda gotta accept that. That’s kind of like asking, why is a number multiplied by 2 always going to be even?
We have found a way to demonstrate how different equations can relate to one another (derivatives and integrals). One of the ways derivatives relate is a function of where the equation you’re deriving *from* falls on a graph.
Take speed and its derivative, distance. Plot speed on a graph. Doesn’t even matter how the line looks, just as long as it’s speed on the y-axis and time on the x-axis.
For any snapshot on that graph, look at the Y-axis value. You see that value is “6 m/s”. So for this snapshot of 1 second on the X-axis, the Y axis says 6 m/s. SO, in that one second, how *far* did the object go? **6 m**.
Because the Y axis is meters/second, and the derivative of meters/sec is meters, then that 6 can demonstrate both the speed AND the derivative of that speed. So if you add ALL the speeds together (area under the graph), then you’re just adding up all the snapshots like the one I just did above.
I’ll just expand this out.
0sec-1sec: 6 m/s
1sec-2sec: 7 m/s
2sec-3sec: 8 m/s
How far did you go? 21 meters. If you plotted that and took the area underneath that 3 second graph of speed, you’d get 21. (Ignoring that you can’t instantaneously change speed)
Imagine you’re walking, then you’re running, then a bit jogging. Integral is the total distance covered.
The rate of change *is* the derivative of a function outside of pathological cases. The area under the curve is the standard integral of the function. If you filled in with a crayon the full area under it, and tried to calculate that area, you’d end up with the integral or anti-derivative.
Derivatives can be described with velocity and displacement.
Derivative of velocity is how fast it’s changing its speed a nd it’s integral would be the distance covered in that time period..
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