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# Definite integral as the limit of a Riemann sum

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.B (LO)
,
LIM‑5.B.1 (EK)
,
LIM‑5.B.2 (EK)
,
LIM‑5.C (LO)
,
LIM‑5.C.1 (EK)
,
LIM‑5.C.2 (EK)

## Video transcript

we've done several videos already where we're approximating the area under a curve by breaking up that area into rectangles and then finding the sum of those the sum of the areas of those rectangles as an approximation and this was just this was actually the first example that we looked at where each of the rectangles had an equal width so we equally partitioned to the interval between our two boundaries between a and B and the height of the rectangle was the function evaluated at the left endpoint of each rectangle and we wanted to generalize it and write it in Sigma notation it looked something like this and this was one case later on we looked at a situation where you define the height by the function value at the right endpoint or at the midpoint and then we even constructed trapezoids and these are all particular instances of Riemann sums Riemann sums so this right over here is a Riemann Riemann sum and when people talk about Riemann sums they're talking about the more general notion you don't have to just do it this way you could use trapezoids you don't even have to have equally spaced partitions I used equally spaced partitions because it made things a little bit conceptually simpler and this right here is a picture of the person that Riemann sums was named after this is bernhard riemann and he did many he made many contributions to mathematics but what he's most known for at least if you're taking a first-year calculus course is the Riemann sum and how this is used to define the Riemann integral both Newton and Leibniz had come up with the idea of the integral when they had formulated calculus but the Riemann integral is kind of the most mainstream formal or I would say rigorous definition of what an integral is so as you can imagine this is one instance of a Riemann sum we have n right over here if the more the larger n is the better and approximation it's going to be so his definition of an integral which is the actual area under the curve or his definition of a definite integral which is the actual area under a curve between a and B is to take this Riemann sum to take this Riemann sum it doesn't have to be this one it take any Riemann sum and take the limit the limit as n approaches infinity so just to be clear what's happening when n approaches infinity let me draw another Grahame here so let's say that's my y-axis this is my x-axis this is my function as n approaches infinity so this is a this is B you're just going to have a ton of rectangles you're just going to get a ton of rectangles over there and they're going to become better and better approximations for the actual area and the actual area under the curve is donated is denoted by the integral the integral from A to B of f of x times DX and just to make it you see where this is coming from or how these notations are closer at least in my brain how they're connected Delta X was the date was the distance for each of these was the width for each of these sections this right here is Delta X so that is a Delta X this is another Delta X this is another Delta X so a reasonable way to conceptualize what DX is or what a differential is is this what Delta X approaches if it becomes infinitely small so you can view this you can conceptualize this and it's not a very rigorous way of thinking about it is an infinitely small infinite infinitely but not 0 infinitely small infinitely small small Delta X is one way is one way that you can conceptualize this so once again if you have your function times a little small change in Delta X and you are summing although you're summing an infinite number of these things from A to B so I'm gonna leave you there just so that you see the connection you know the name for these things and once again this one over here this isn't this isn't the only Riemann sum in fact this is often called the left Riemann sum if you're using it with rectangles you can do a right Riemann sum you can do use the midpoint you could use a trapezoid but if you take the limit of any of those Riemann sums as n approaches infinity then that you get as a as a riemann definition of the integral now so far we haven't we haven't talked about how to actually evaluate this thing this is just a definition right now and for that we will do in future videos
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