Worked example: Rewriting limit of Riemann sum as definite integral

- [Instructor] So, we've got a Riemann sum. We're gonna take the limit as N approaches infinity and the goal of this video is to see if we can rewrite this as a definite integral. I encourage you to pause the video and see if you can work through it on your own. So, let's remind ourselves how a definite integral can relate to a Riemann sum. So, if I have the definite integral from A to B of F of X, F of X, DX, we have seen in other videos this is going to be the limit as N approaches infinity of the sum, capital sigma, going from I equals one to N and so, essentially we're gonna sum the areas of a bunch of rectangles where the width of each of those rectangles we can write as a delta X, so your width is going to be delta X of each of those rectangles and then your height is going to be the value of the function evaluated some place in that delta X. If we're doing a right Riemann sum we would do the right end of that rectangle or of that sub interval and so, we would start at our lower bound A and we would add as many delta Xs as our index specifies. So, if I is equal to one, we add one delta X, so we would be at the right of the first rectangle. If I is equal two, we add two delta Xs. So, this is going to be delta X times our index. So, this is the general form that we have seen before and so, one possibility, you could even do a little bit of pattern matching right here, our function looks like the natural log function, so that looks like our func F of X, it's the natural log function, so I could write that, so F of X looks like the natural log of X. What else do we see? Well, A, that looks like two. A is equal to two. What would our delta X be? Well, you can see this right over here, this thing that we're multiplying that just is divided by N and it's not multiplying by an I, this looks like our delta X and this right over here looks like delta X times I. So, it looks like our delta X is equal to five over N. So, what can we tell so far? Well, we could say that, okay, this thing up here, up the original thing is going to be equal to the definite integral, we know our lower bound is going from two to we haven't figured out our upper bound yet, we haven't figured out our B yet but our function is the natural log of X and then I will just write a DX here. So, in order to complete writing this definite integral I need to be able to write the upper bound and the way to figure out the upper bound is by looking at our delta X because the way that we would figure out a delta X for this Riemann sum here, we would say that delta X is equal to the difference between our bounds divided by how many sections we want to divide it in, divided by N. So, it's equals to B minus A, B minus A over N, over N and so, you can pattern match here. If this is delta X is equal to B minus A over N. Let me write this down. So, this is going to be equal to B, B minus our A which is two, all of that over N, so B minus two is equal to five which would make B equal to seven. B is equal to seven. So, there you have it. We have our original limit, our Riemann limit or our limit of our Riemann sum being rewritten as a definite integral. And once again, I want to emphasize why this makes sense. If we wanted to draw this it would look something like this, I'm gonna try to hand draw the natural log function, it looks something like this and this right over here would be one and so, let's say this is two and so going from two to seven, this isn't exactly right and so, our definite integral is concerned with the area under the curve from two until seven and so, this Riemann sum you can view as an approximation when N isn't approaching infinity but what you're saying is look, when I is equal to one, your first one is going to be of width five over N, so this is essentially saying our difference between two and seven, we're taking that distance five, dividing it into N rectangles, and so, this first one is going to have a width of five over N and then what's the height gonna be? Well, it's a right Riemann sum, so we're using the value of the function right over there, write it two plus five over N. So, this value right over here. This is the natural log, the natural log of two plus five over N, and since this is the first rectangle times one, times one. Now we could keep going. This one right over here the width is the same, five over N but what's the height? Well, the height here, this height right over here is going to be the natural log of two plus five over N times two, times two. This is for I is equal to two. This is I is equal to one. And so, hopefully you are seeing that this makes sense. The area of this first rectangle is going to be natural log of two plus five over N times one times five over N and the second one over here, natural log of two plus five over N times two times five over N and so, this is calculating the sum of the areas of these rectangles but then it's taking the limit as N approaches infinity so we get better and better approximations going all the way to the exact area.