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# Interpreting definite integral as net change

AP.CALC:
CHA‑4 (EU)
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CHA‑4.D (LO)
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CHA‑4.D.1 (EK)
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CHA‑4.D.2 (EK)
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CHA‑4.E (LO)
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CHA‑4.E.1 (EK)

## Video transcript

in a previous video we start to get an intuition for rate curves and what the area under a rate curve represents so for example this rate curve this might represent the speed of a car and how the speed of a car is changing with respect to time and so this shows us that our rate is actually changing this isn't distance as a function of time this is rate as a function of time so this looks like the car is accelerating at time one it is going 10 meters per second and at time 5 let's say let's assume that all of these are in seconds so at 5 seconds it is going 20 meters per second so it is accelerating now the relationship between the rate function and the area is that if we're able to figure out this area then that is the change in distance of the car so rate or speed in this case is distance per unit time if we're able to figure out the area under that curve it will actually give us our change in distance from time 1 to time 5 it won't tell us our total distance because we don't we won't know what happened before time 1 if we're not concerned with that area and the intuition for that it's a little bit easier if you were dealing with rectangles but just think about this let's make a rectangle that looks like a pretty good approximation for the area let's say from time one to time two right over here well what is this area of this rectangle represent we would to figure out the area we would multiply one second that would be the width here times roughly it looks like about 10 meters per second and so the units here would be 10 meters per second times 1 second or 10 meters and we know from early physics or even before that if you multiply a rate times time or a speed times a time you're going to get a distance and so the unit here is in distance and as you can see this area is going to represent its are gonna be an approximation for the distance traveled and so if you wanted to get an exact version or an exact number for the distance traveled you would get the exact area under the curve and we have a notation for that if you want the exact area under the curve right over here we use definite integral notation this area right over here we could denote as a definite integral from 1 to 5 of R of T DT and once again what does this represent in this case when our rate is speed this represents this whole expression represents our change in distance from T is equal to 1 to T is equal to 5 now with that context let's actually try to do an example problem the type that you might see on Khan Academy so this right over here tells us even walked at a rate of R of T kilometers per hour where T is the time in hours okay so now T is in hours what does the integral from the definite integral from 2 to 3 of our of T DT equals 6 mean so before I even look at these choices this is saying so this is going from T equals two hours to T equals three hours and it's essentially the area under the rate curve and here the rate is we're talking about a speed Eden is walking at a certain number of kilometres per hour so what this means is that from time to hours to time three hours Eden walked an extra six kilometers so let's see which of these choices match that Eden walked six kilometers each hour it does tell us that from time to two three Eden walked six kilometers but doesn't just mean but we don't know what happened from time 0 to time 1 or from time one to time two so I would rule this out Eden walked 6 kilometers in three hours so this is a common misconception people will look at the top bound and say okay the this is this area represent by the definite integral this tells us how far in total we have walked up until that point that is not what this represents this represents the change in distance from time - to time 3 so I'll rule that out edan walked six kilometers during the third hour yes that's what we've been talking about from time to equals hours from time equal to hours to time equal three hours edan walk six kilometers and you could view that as the third hour going from time to to time three Eden's rate increased by six kilometers per hour but between hours 2 & 3 so let's be very clear this right over here this isn't a rate this is the area under the rate curve that's what this definite integral is representing and so this isn't telling us about our rate changing this is telling us how does the thing that the rate is measuring the change of how does that change from time to time 3 so we would rule that out as well
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