 Hey everyone, I wanna do one more example from section 5.1 from Stuart's textbook here about finding the area under the curve. And one, I wanna give you some applications of why we are interested in this area problem in the first place. And also two, I want to kind of show you that one doesn't need to have an algebraic function to do these things. I mean those calculators we used before were nice but we can do these things without such a calculator. And so to kind of help us out here, think of the following. Suppose we have a driver who's traveling on a business trip and he checks this pedometer every hour. And so the table here below shows the speed or the velocity, right? So we check the different times, zero hours, one hour, two hour, three hours. And we can check the velocity zero, zero miles per hour, 52 for the eight and 60. And so we wanna approximate the total distance that the driver goes during this three hour period. And the thing we have is essentially the following. We only know a little bit about the function. We know what happens at zero, one, two and three of this velocity function. And we know it's gonna be here zero, 52, 58 and 60. And that's all we know about the function. We just have these three dots. How do we estimate the area under the curve? That is how can we approximate the total distance? Well, the idea is if we know what the average speed is for the first hour, we can then estimate how far they went. Like if he was driving 80 miles per hour in the first hour, he would have gone 80 miles. So we wanna sort of estimate what's the speed that the driver was doing in those timeframes. And so this comes down to approximate the area under the curve right here. So let's say we first try to calculate the area using the left endpoint rule. Well, there's three hours, so we could do L3 right here, which to be aware, since we know there's three hours, our delta X, which would be three minus zero. The best we could subdivide it would be by three hours because we don't have any more refined data than that. So we can do a one hour increment here. So what we can do is we can use the left end points where you use zero for the first interval. You're gonna use 58 for the second one and then you're gonna use the fifth, sorry, 52 for the second one and 58 for the third one. So you're gonna get 52 times one, 58 times one and zero times one. And so L3 here, you're gonna get one times zero plus 52 plus 58 like so. That'll add up to be 110 miles. Because what you're doing here is like we're estimating, okay, the first hour, the person drove zero miles per hour. He did that for an hour, so he went zero miles. For the second hour, he was going on average speed of 52 miles per hour. You times that by one, he went 52 miles. That's what our estimate is over here. On the other hand, if we use the right point rule, R3, we determine the height of the rectangles by the right end point and then we get the right point right here and then using the right point, we get this right here. So for the first one, we would estimate that speed to be 52 miles per hour. For the second one, we'll get the 58 and for the third one, we're gonna get 60. The delta X will still be the same in this situation. We get 52, plus 58, plus 60. And so this gives us 170 this time. 170 miles, we expected. Now, the lower estimate is probably, the left rule is probably underestimating this thing. I mean, after all, zero miles for the first hour, he just didn't do anything, I doubt that. And R3 might be overestimating things. It's hard to say with this data here. We can't do any more refined than we are. But we can try to estimate, we can estimate, even somewhere between 110 miles, 170. If we do the trapezoid rule, T3 will take L3 plus R3 over two, which is 110 plus 170 over two. Right? Let's see, do I have any space here? You're gonna get 280 over two. And once you cut that in half, you end up with 140 miles would be the estimated there. And so even with this tiny little bit of data, we can get a pretty good estimate of how far our driver went on this business trip. So approximately 140 miles. And so from an illustrative point of view, these are the types of estimates under the curve I want students to be able to do. Use the calculators when appropriate, but come exam time, if you don't have a calculator like that, I do want you to understand these basic calculations, just like on a small example like this.