Word problems involving definite integrals
Interpreting definite integral as net change
- [Instructor] In a previous video we started 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 a speed of a car and how a 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 function of time, this is rate as a function of time. So, this looks like car is accelerating. At time one it is going ten meters per second and at time five, let's assume that all of these are in seconds, so at five seconds, it is going 20 meters per second. So, it is 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 one to time five. It won't tell us our total distance, cause we won't know what happens before time one, if we're not concerned with that area. And the intuition for that, it's a little bit easier if you're 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 from the rectangle represent? To figure out the area we would multiply one second, that would be the width here, times roughly, looks like about ten meters per second. And so the units here would be ten meters per second, times on second, or ten 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 gonna get a distance. And so the unit here is in distance, as you can see, this area is going to represent, it's 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 can denote as a definite integral from one to five, 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 reprsents our change in distance from t is equal to one, to t is equal to five. 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, Eden walked at a rate of r of t kilometers per hour, where t is the time in hours. Okay, so t is in hours. What does the integral, the definite integral from two to three of r of t dt equals six 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 kilometers per hour. So what this means, that from time two 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 from time two to three, Eden walked six kilometers, but it doesn't mean, but we don't know what happened from time zero to time one, or from time one to time two. So I would rule this out. Eden walked six kilometers in three hours. So this is a common misconception, people will look at the top bound and say okay, this area represented 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 two to time three. So, I'll rule that out. Eden walked six kilometers during the third hour. Yes, that's what we've been talking about. From time equal two hours to time equal three hours, Eden walked six kilometers and you could view that as the third hour, going from time two to time three. Eden's rate increased by six kilometers per hour between hours two and three. So let's be very clear, this right over here, this isn't a rate, this is the area under the rate curve, this is 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 two to time three? So we would rule that out as well.