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# Worked example: Rewriting definite integral as limit of 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

let's get some practice rewriting definite integrals as the limit of a Riemann sum so let's say I wanted to take the definite integral from PI to 2 pi of cosine of X DX and what I want to do is I want to write it as the limit as n approaches infinity of a Riemann sum so we're going to take the form of the limit as n approaches infinity and we can have our Sigma notation right over here and I would say from let's say I is equal to 1 all the way to n let's scroll down a little bit so it doesn't get all scrunched up at the top of it so let me draw what's actually going on so that we can get a better sense of what to write here within the Sigma notation so large so if this is pi right over here this would be 3 PI over 2 and this would be 2 pi all right over here 2 pi now what is the graph of cosine of X do well at PI cosine of PI is negative 1 well so that's negative 1 there and cosine of 2 pi is 1 and so the graph is going to do something like this and I'm just as obviously just a hand-drawn version of it but you have seen cosine functions before this is just part of it and so this definite integral represents the area from PI to 2 pi between the curve and the x-axis and you might already know that this area is going to be or this depth this part of the definite integral would be negative and this would be positive and will cancel out and this would all actually end up being 0 in this case but this exercise for this video is to rewrite this in re as the limit as n approaches infinity of a Riemann sum so there's a Riemann sum what we want to do is think about breaking this up into a bunch of rectangles so let's say Aurel I should say and rectangles so that's our first one right over there then this might be our second one let's do right Riemann sum where the right boundary of our rectangle what the value of the function is at that point that's what defines the height so that's our second one all the way until this one right over here is going to be our nth one so this is one let me write it this way this is I is equal to 1 this is I is equal to two all the way until we get to I is equal to n and then if we take the limit as n approaches infinity the sum of the areas of these rectangles are going to get better and better and better and so let's first think about it what is the width of each of these rectangles going to be well I am taking this interval from PI to 2 pi and I'm going to divide it into n equal intervals so the width of each of these the width of each of these is going to be 2 pi minus pi so I'm just taking the difference between my bounds of integration and I am dividing by n which is equal to PI over n so that's the width of each of these that's PI over N this is PI over N this is PI over N and what's the height of each of these rectangles remember this is a right Riemann sum so it's going to be the right end of our bow of our rectangle is going to define the height so this right over here what would this height be well this height this value I should say this is going to be equal to F of what well this was PI and this is going to be PI plus this the length of our interval right over here the base of the rectangle so we started at PI so it's going to be PI plus this one's going to be PI over n I could say times 1 that's this height right over here what's this one going to be right over here well this one is going to be F of Pi our first start plus PI over n times what we're going to add PI over n 2 times pi over n times 2 so the general form of the right boundary is going to be so for example this height right over here this is going to be f of we started at PI plus we're doing the right Riemann sum so we're going to add PI over N n times by this point PI over N times n or if we wanted to say generally if we're talking about the I to rectangle remember going to sum them all up what's their height well the height is going to be in this case it's going to be cosine of pi plus if we're with the height or rectangle we are going to add PI over n I times pi over n times I and then that's the height of each of our rectangles and then what's the width well we already figure that out times pi over N and if you want to be careful and make sure that this Sigma notation applies to the whole thing there you have it we have just reexpress to this definite integral as the limit of a right Riemann sum
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