Area under rate function gives the net change

let's say that something is traveling at a constant rate of five meters per second that's its velocity in one dimension if it was negative would be moving to the left if it's positive it's moving to the right and let's say that we we care about what is our change in what is our change in distance over would this the Delta symbol represents change over a change in time of four seconds over four seconds and I could say from T equals zero to T is equal to four that's our change in time that's our four second interval that we care about well one way to think about it is well a rate a rate by definition is nothing but a change in some quantity in this case its distance over over a change in some other quantity in this case we're thinking about time or another way to think about it if we multiply both time both sides times change in time you get your change in distance is equal to your rate is equal to your rate times change in time rate times change in time and so this is very close to you might remember from pre-algebra distance is equal to rate times time and that just comes from in the definition of what a rate is it's a change in one quantity with respect to another quantity and so if you just apply this if you say okay my rate is a constant five meters per second and my delta T is I'm as four seconds so times four seconds well that's just going to give you 20 that's just going to get you 20 let me do that same color that I had for the for the change in distance that's going to be 20 and then the seconds cancel with seconds 20 meters so my total change in distance over those four seconds is going to be 20 meters nothing new here nothing too fancy what I want to do now is connect this to the area under the rate function over this time period so let's graph that let's graph it so that's my rate axis this is my time axis this is going to be in seconds this is going to be in meters per second so let's see one two three four five that seems about enough and then I go one two three four five our rate at least in this example is a constant is a constant five meters per second it's a constant five meters per second so that is my R of T in this example and so what did we just do here we just multiplied our change in time times our constant rate we just multiplied our change in time so it was from time equals zero seconds to four seconds so it's this length here if we think of it on that axis and we multiplied it times our constant rate we multiplied it times this right over here well if I multiply this base times this height what am I going to get I'm going to get the area under I am going to get this area under the rate function and that area is going to be 20 and if we went with the units of them obviously you're used to things of area being something you know units squared because it's usually you know meters times meters or miles times miles or inches times inches would be inches squared meters squared or mile squared but here if we go with the units of the axes it would be meters per second times seconds which is going to get you meters but the important thing here is that the units here or the area here is 20 so least for that very simple example it looks like the area under the rate curve is equal to the net change over that time period where the rate is something with respect where we're saying that the rate of something with respect to time but let's test that a little bar let's just get a little more intuition here let's say that we had a different rate function let's say that let me make it with a different let's say instead we had a rate function I'll still use yellow let's say I have a rate function that is let me make it a little bit interesting let's say it's one meter per second for our time is greater than or equal or zero is less than or equal to time which is less than or equal to two seconds and and this is obviously this is all in seconds when we're talking about time and there's two meters per second 40 is greater than two seconds so what's that going to look like and actually try to graph it yourself and just say well what is the total change in distance over the first let's say five seconds so we want to do delta T over the first four seconds but the first five seconds well let's graph it let's graph it so this is one meter per second one meter per second that is two meters per second it's in meters per second that's my rate axis and and this right over there is going to be my that's going to be my time axis one they're obviously not at the same scale three four five and what is this what does this rate function look like well my rate is one meter per second between time is zero and two including two seconds and then the rate jump so this isn't that realistic nothing can accelerate instantly like this you would need an infinite force or an infinitely small mass I guess to well maybe there's some things if we think about well anyway I don't get too complex there but it this is unrealistic more but it's not typical for something to to just have an instantaneous velocity increase like that but let's just go with it so so then after the two seconds we are at a constant we are at a constant rate of two meters per second now what is our total change in distance over the first five seconds here so here we care about the first five seconds well we can break up the problem we could say well over the first two seconds change in time is two seconds times R we have a constant rate over those two seconds so it's going to be two seconds times one meter per second well that's going to be that's going to give us two meters so this here is going to be actually let me do that orange color that's going to give us two meters there and then we look at the next section our change in time here is three seconds and then we multiply that times our constant to me per second that's going to give us an area of six and if we look at the units in both cases we're multiplying seconds times meters per second which is going to give us meters so it's going to be 2 plus 6 meters or 8 meters so hopefully this is giving you the intuition that the area under the rate curve or the rate function is going to give you our total net change in whatever that rate thing was it was finding the rate of in this case it is distance per unit time if you if you take the area under the rate function that that kind of distance per this the speed or this velocity function over some period of time that area is going to be our total net change in distance