![]() Going to have a width of five over N and then what's Saying our difference between two and seven, we're taking that distance five, dividing it into N rectangles, and so, this first one is Isn't approaching infinity but what you're saying is look, when I is equal to one, your first one is going toīe of width five over N, so this is essentially 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 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 If we wanted to draw this it would look something like this, I'm gonna try to hand draw And once again, I want to emphasize why this makes sense. ![]() Limit, our Riemann limit or our limit of our Riemann sum being rewritten as a definite integral. 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. ![]() If this is delta X isĮqual to B minus A over N. So, it's equals to B minus A, B minus A over N, over N and so, you can pattern match here. Is equal to the difference between our bounds dividedīy how many sections we want to divide it in, divided by N. Would figure out a delta X for this Riemann sum here, we would say that delta X 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 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. Is going to be equal to the definite integral, we know our lower bound is going from two to we haven't figured Okay, this thing up here, up the original thing So, what can we tell so far? Well, we could say that, 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 What else do we see? Well, A, that looks like two. 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. So, this is the general form that we have seen before and so, one possibility, youĬould even do a little bit of pattern matching right here, our function looks like So, this is going to be delta X times our index. So, if I is equal to one, we add one delta X, so we would be at the right If we're doing a right Riemann sum we would do the rightĮnd of that rectangle or of that sub interval and so, we would startĪt our lower bound A and we would add as many delta 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. We're gonna sum the areas of a bunch of rectangles where the width of each Of the sum, capital sigma, going from I equals one to N and so, essentially 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 So, let's remind ourselves how a definite integral can I encourage you to pause the video and see if you can work ![]() We're gonna take the limitĪs N approaches infinity and the goal of this video is to see if we can rewrite this as a definite integral.
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