 This lesson is on how do we measure distance traveled. We are going to use velocity versus time in order to determine distance. To do that, I have an example for you. How far did I travel? My trip to North Carolina School of Science and Math takes about 40 minutes each day, each way. I wrote down the velocity I was traveling at every five minutes, and this is what I got. Time was in minutes, and velocity is in miles per hour. When I started off, of course, I was going zero miles per hour. Five minutes later, I was going about 25. Ten minutes later, about 60 miles an hour. Fifteen minutes later, I was still going 60 miles an hour. Twenty minutes later, I was at a light, and that was stopped, of course, red, zero. At 25, I was going 50 miles an hour. 30 minutes, 60 miles an hour. At 35 minutes, I was going 35 miles an hour. At 40 minutes, I wasn't quite at school, but I was almost here, and I had been slowing down to about 15 miles an hour. How far did I travel? Well, in calculus, we have several ways of figuring this out, and one of the ways we are going to work on today is called rectangular approximation method. And you realize that time, times velocity, equals distance. So we're going to use that idea when we create our rectangles and approximate how far I traveled. The first type we are going to use is called left rectangular approximation method. And in order to show you how to do this, I have plotted out all the points of my velocity as I was traveling along to school. So this is a time, which was in minutes, and velocity graph, which is miles per hour. And of course, when I actually compute, I have to convert. So if we create rectangles from these different points, and I create the left point to be the height of my rectangle, the first velocity I had when the time was zero was the velocity was zero. So that actually has no height for that rectangle. The second one, I was going about 25 miles an hour. So that height is around 25. The third time I measured, I was going 60 miles an hour. So I have a height of 60 on this rectangle. The next time I measured, I was going 60 miles an hour. So I have that height. And then I slowed down to zero miles per hour. So that's a zero back up to 50. So that one has a height of 50. The next one has a height of 60 miles per hour. The next one is 35 miles per hour. And that creates our last rectangle. So this point here is not used at all when we are doing L-RAMs. So in order to find the distance, I know each increment is five minutes, or five out of 60, because my distance is in miles per hour, and I have to create hour here. So it's five minutes out of 60 minutes, which is one-twelfth of an hour. And then each of the heights can be added together because every increment along the time is the same. So I can do five over 60 there and just multiply by the heights. So this one was zero. The next one is 25. The next one is 60. Another 60, a zero, a 50, a 60, and a 35. And if I add all of those together and multiply it by one-twelfth, I should get 24.167 miles. So during my 40 minutes, I traveled approximately 24 miles, doing it left rectangular approximation method. Now if we do the same thing with the right rectangular approximation, instead of using a left point as the height of my rectangle, I'm going to use right points. And this is how this one looks. So when I make each rectangle, instead of using the zero, I'm using the 25. And this one, I'm using the 60. So that height is 60. Again, using a 60. Then to the zero, for that right hand, and then up to 50, for that right hand. And then 60. Of course, most of the numbers are being used except for the two end ones. They change depending on whether you're using right or left rectangular approximation method. So this one is 35. And the last one is 15. So on this one, we're doing r-gram. If again, I try to find the distance, I do the 5 over 60. And instead of starting with zero, I start with 25. And then add the two 60s in. Add the zero, the 50, 60, 35. And the last one has to be put in, which is 15. And this one should be higher because if you remember, when I did left rectangular approximation method, I started off with a zero. Well, that zero is gone and said I have a 15 sort of in its place. So this number should be higher, which it is. It's approximately 25.417 miles. So we have our left rectangular approximation method, our right rectangular approximation method. And of course, one gives a lower estimate, as you can see. One gives a higher estimate. We have one that is a little bit more accurate. And that's called M or mid-rectangular approximation method. And the way we do that one is to take the average between the two points on either side of the interval in question of that rectangle we're creating. So if you remember the first point is zero, zero. The next one is five, 25. So halfway between zero and 25 is 12 and a half. So we create a rectangle at 12.5. And then we go from 25 to 60. And the height of that rectangle is halfway between this point and this point, which is about here, which is 42.5. Then we have our two sixties. And of course, if you average that out, it becomes still 60. And then we go from 60 to zero in our velocity. So the height of that rectangle is halfway between, which is at 30. Then we go from zero to 50. So halfway between is 25. And then we go from 50 to 60, which is at 55. We take that average. And then we go from 60 to 35 miles per hour. So halfway between is 47.5. And the last one we go from 35 to 15 halfway between is 25. And those are the heights we are going to use for our rectangular approximation method. So our distance here is five over 60 again, times 12.5 plus 42.5 plus 60, plus 30, plus 25, plus 55, plus 47.5 plus 25. And that gives us approximately 24.797 miles. Now, if you remember, our L-RAM was 24.167 miles. The R-RAM was 25.417 miles. And this one is in between at 24.7917 miles. Well, what happens if I decide to take my readings every 2.5 minutes instead of every 5 minutes? Well, you'll see my rectangles get closer and closer together. Would I get a better estimate of my distance? One would think so. And I actually wrote those down and did them. And you may try them if you want to practice this. But you will see that the closer those rectangles are together or the closer I take my times, the more accurate my answer should be. Well, let's look at the theory behind all this. We have the left-hand sums and the right-hand sums we want to look at here. If I were to take this as my beginning point on a graph, and this is my end point on the graph A to B, and let's say this is a time velocity graph, which we're going to measure distance. This is our T sub 0, which means the height is V sub 0. And the next point is T sub 1. And then the next one will be T sub 2. And we'll continue on until we get to B. And there will be many of these drawn in between. And we say B is the T sub n, which makes that height V sub n. Well, what will the sum look like? Well, our new distance will be whatever increment we use, which is the delta T. And delta T is found by determining B minus A over the number of subdivisions that we make. And then we will multiply this by each of the heights that we have. So if we were doing left RAM, we would start out at A or T sub 0, which would give us a height of V sub 0. And then we would add V sub 1 plus V sub 2 all the way up to not V sub n. Because remember, we don't use the last one, we use the one before V sub n minus 1. And that will give us the distance if we use L RAM. If we want our RAM, again, we would have the delta T. And instead of starting at V sub 0, we will now start where T sub 1 meets V sub 1. So we would start at the V sub 1, add the V sub 2, add the V sub 3, all the way to the end, including the last one, V sub n. That is the theory behind this. And we will be developing this theory more as we go into the next lesson. This concludes our lesson on how do I measure distance traveled.