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Current time:0:00Total duration:7:07

Worked example: Riemann sums in summation notation

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

Video transcript

what I want to do in this video is get a little bit of practice trying to approximate the area under curves and also get a little bit more familiarity with Sigma notation in this context so what we have here we have the graph of f of X is equal to 1 plus 0.1 x squared that's this curve right over here and then we have these rectangles that are trying to approximate the area under the curve the area under the function f between x equals 0 and x equals 8 and the way that this diagram or the way that we're attempting to do it is by splitting it into four rectangles and so we could call this rectangle 1 this is rectangle 2 rectangle 3 and rectangle 4 and each of their heights let's see they're the interval it looks like they each have a width of 2 so they are equally spaced so we go from 0 to 8 and we've split it into 4 sections so each has a width of 2 so they're each going to be too wide so that's two that's two that's two that's two and their height seems to be based on theirs height seems to be based on the midpoint so between the start between the left side and the right side of the rectangle you take the value of the function at at the middle value right over here so for example this height right over here looks like f of 1 this height right over here looks like f of 3 this height of this rectangle is f of 5 this height right over here is f of 7 so given the way that this has been constructed and we want to take the sum of the areas of these rectangles as an approximation under as the area of this curve how would we write that as Sigma notation and I'm going to start it and then I encourage you to pause the video and try to finish it so the the some of these the some of these rectangles we could say it's the sum of so we'll have N equals 1 to 4 because we have for rectangular and I encourage you to and I encourage you to to finish this up once actually just write it in terms of the function use function notation you don't have to write it out as 1 plus 0.01 something squared so I'm assuming you've had a go at it so for each of these so for the first so for the first rectangle over here we're going to multiply two times the height so the height right over here is one and this is the first rectangle so you might be tempted to say times f of n but then that breaks down as we go into the second rectangle the second rectangle the two still applies this to is the width of the rectangle but now we want to multiply it times F of three not F of two so this F of n isn't going to this F of n isn't going to isn't going to pass muster and so let's see let's see how we want to think about it so when n is so when n is one two three four we're going to take f of so F of n or exactly I should say F of n we're going to take F of something so here this first one we're going to take F of one then here for the second rectangle we're taking F of three F of 3 for the height for the third rectangle we're taking F of five F of five and then so for the fourth rectangle we're taking F of seven F of seven so what's the relationship over here let's see it looks like if you multiply by 2 and subtract one so 2 times 1 minus 1 is 1/2 times 2 minus 1 is 3 2 times 3 minus 5 2 times 3 minus 1 is 5 2 times 4 minus 1 is 7 so this is 2 n minus 1 so the area of each of these rectangles the base is 2 and the height is f of 2 n minus F of 2 n minus 1 and so that's that hopefully makes a little bit clearer kind of mapping between the Sigma notation and and what we're actually trying to do and now it's just just for fun let's actually try to evaluate this thing what is this thing going to evaluate to well this is going to evaluate to 2 times F of when n is equal to 1 this is 1 F of 1 plus 2 aimes when n is 2 is going to be F of 2 times 2 minus 1 is 3 F of 3 but when n is 3 this is going to be 2 times F of 5 when n is 4 this is going to be 2 times 2 times F of 7 4 times 2 minus 1 is 7 F of 7 and so that is going to be we're going to have to evaluate a bunch of these things over here so let me actually let me clear this out so I have a little bit more real estate feeling this might get a little bit messy now so this is going to be actually we could factor out a 2 so this is going to be equal to 2 times F of 1 is 1 plus 0.1 times 1 squared so it's 1 plus 0.1 so it's one point let me color code it a little bit just so we could keep track of things so this right over here is 1 point 1 so z1 plus 0.1 is 1 point 1 this right over here F of 3 so this one plus point 1 times 3 squared 9 so 1 plus 0.9 so 1 plus so it's 1 point 9 and then let's see this one right over here F of 5 that's going to be 1 plus C 5 squared is 25 times 0.1 is 2 point 5 so 1 plus 2 point 5 is going to be 3 point 5 and then finally F 1 finally F of 7 F of 7 is going to be 1 plus point 1 times 7 squared so this is 49 times 0.1 49 times point one is four point nine plus one so plus five point nine plus five point nine and so what is this going to be equal to so let's see one point one plus one point nine these two are going to sum up to be equal to three and then these two are going to sum up to be let's see if we take if we add the 5 we get to eight point five and then we add the point 9 we get to nine point four so plus nine point four did I do that right three plus five is 85.5 plus 0.9 is 1.4 yep and so this is going to be so once again we have the two times at all so this is going to be equal to two times twelve point four which is equal to twenty four point eight which is our approximation once again this is just an approximation using these rectangles of the area under the curve between x equals zero and x equals eight