 Hi, and welcome to the screencast on estimating distance traveled from a velocity graph. As the title suggests, in this screencast we're going to practice the basic technique for estimating the distance traveled by a moving object if we're given a graph of its velocity. The specific example here will use this curve that shows the velocity in meters per second of a moving object over a five second interval. As you can see, the velocity is not constant. There's an overall slowing down trend in the velocity, but the velocity also surges up and down as the object slows like a car pumping its brakes as it comes close to a stop sign. We're going to estimate the distance traveled by this object over the interval from zero seconds to five seconds. Now the overall concept here goes like this. First, we're going to subdivide the five second time interval into equally sized pieces. Then we will form rectangles over each subdivision with a height equal to the height of the left-hand endpoint. Then we'll form the area in each rectangle and find it using basic geometry. Finally, we'll add up all those areas and this will give us an estimate of the distance traveled. So let's start with a very crude estimate using just one big rectangle. For this estimate, I'm not going to subdivide the time interval at all but just treat it as a single five second long time span. To form the one rectangle that results from this, I'm going to go to the left-hand endpoint here at time equals zero and send that point up to the curve. Here's the rectangle that results. Now let's find the area. Now the area of a rectangle, as you know, is base length times height. The base length here is five and the height appears to be ten. So the area is 50. Now this is an estimate to the distance traveled because although this is an area, let's take a look at the units. The base is measured in seconds because it's a time span. The height is measured in meters per second because it's a velocity. So the product of those two is meters per second times seconds, which comes out to meters and that's a distance traveled. What we've done here is use the old rule that distance equals rate times time. We're assuming here that the object is traveling at a constant ten meters per second for the entire five second period. So that lets us use the distance equals rate times time formula and gives us an estimated distance traveled of 50 meters. Obviously though, this is a bad estimate because in fact, the object is definitely not traveling at ten meters per second the whole time. In fact, it would appear that the distance traveled is a lot less, maybe as much as 50% less than the 50 meters that we estimated. So we need to improve upon this estimate. We can do that by subdividing the time interval into smaller pieces. Let's say we wanted to subdivide into two equal pieces. That would mean I'd have one sub-interval that goes from zero seconds to two and a half seconds, and another sub-interval that goes from two and a half seconds to five seconds. In each of those sub-intervals, let's go to the left hand endpoint and direct a rectangle from those points to the curve. Now let's look at the area of each rectangle. The area of the first one is 10 by 2.5, which is 25. The area of the second is about five and a quarter times a width of two and a half, which is about 13.125. Adding these up, we get 25 plus 13.125, which is 38.125. Again, the units on this quality are meters because each rectangle area was calculated by multiplying a rate times a time to get distance. And we can get away with that here because we're assuming that the rate is constant. So this is an improved estimate of the total distance traveled by this object over the five second interval. And you can see that it's improved because there's a lot less extra space that's included by the rectangles this time. That's compared to the first time with one big rectangle. Now we can continue to improve this estimate by clearing off the rectangles and recalculating this area sum using even more rectangles. For example, here's what the picture looks like with five subdivisions. For a quick concept check, pause the video and see if you can calculate the estimated distance traveled with five rectangles as shown here. So let's look at the solution with five rectangles. My estimate may not exactly equal yours because in some cases, we have to visually estimate the height values off the graph. And you and I might not agree 100% on those estimates. But they should be in the ballpark of each other. And as long as you're within a few tenths of a meter here from what I get, then I think we're both okay. Each rectangle in the five rectangle estimate has a width equal to one because we're dividing up a five second interval into five equal pieces. Now let's go to the left hand endpoint of each of these subintervals and look at the heights of each of the rectangles. The height of the first rectangle is at ten. The height of the second one appears to be about nine and a half. The third, about 6.25. The fourth, about 5.25. And the fifth and final one, right around five. Now to calculate the area sum, we're just going to find the areas of each of those rectangles and add them up. Here that would be ten times one plus 9.5 times one plus 6.25 times one plus 5.25 times one plus five times one. That adds up to 36, and that's measured in meters again for reasons we explained above, and that gives us the estimate of the distance traveled. That's all we're going to do in this screencast, but realize that we could make this estimate as accurate as we wish, just by putting in more and more rectangles into the picture. For example, here's what the picture looks like when we use 25 rectangles. And the sum of all those areas in case you're interested is 32.96. And visually this appears to be a very close estimate indeed. However, there is a trade off, of course. If we want more accuracy, then we're going to have to do a lot more computation. In this case, 25 area calculations. That's all for now, thanks for watching.