Interpreting definite integral as net change | AP Calculus AB | Khan Academy

In a previous video, we start to get an intuition for rate curves and what the area under a rate curve represents. For example, this rate curve might represent the speed of a car and how the speed of a car is changing with respect to time. 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 1, it is going 10 meters per second, and at time 5, let's say—let's assume that all of these are in seconds—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 won't know what happened before time 1 if we're not concerned with that area. The intuition for that is 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 1 to time 2 right over here.

Well, what does this area of this rectangle represent? To figure out the area, we'd multiply 1 second—that would be the width here—times roughly it looks like about 10 meters per second. The units here would be 10 meters per second times one second or 10 meters.

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. The unit here is in distance, and as you can see, this area is going to represent—it's going to be an approximation for the distance traveled.

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. 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. Once again, what does this represent in this case? When our rate is speed, 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 Eden 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, the definite integral from two to three of r of t dt equals six mean? 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.

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 is 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 that 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. This is a common misconception. People will look at the top bound and say, "Okay, this is the 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 t equal to two hours to time t equal to three hours, Eden walks 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.

But 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. That's what this definite integral is representing. 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.