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# Over- and under-estimation of Riemann sums

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.A (LO)
,
LIM‑5.A.1 (EK)
,
LIM‑5.A.2 (EK)
,
LIM‑5.A.3 (EK)
,
LIM‑5.A.4 (EK)

## Video transcript

consider the left and right Riemann sums that would approximate the area under y is equal to G of X between x equals 2 and x equals 8 so we want to approximate this light blue area right over here are the approximations over estimations or under estimations so let's just think about each of them let's consider the left and the right Riemann sums so first the left and I'm just gonna write left 4 short but I'm talking about the left Riemann sum so they don't tell us how many subdivisions to make for our approximation so that's up to us to decide let's say we went with three subdivisions let's say we wanted to make them equal they don't have to be but let's say we do so the first one would go from 2 to 4 the next will go from 4 to 6 and the next one go from 6 to 8 if we do a left Riemann sum you use the left side of each of these subdivisions in order to define the height you evaluate the function at the left end of each of those subdivisions for the height of our approximating rectangles so we would use G of 2 to approximate for or to set the height of our first approximating rectangle just like that and then we would use G of 4 for the next rectangle so we would be right over there and then you'd use G of 6 to represent the height of our third and our final rectangle right over there now when it's drawn out like this it's pretty clear that our left Riemann sum is going to be an over estimation why do we know that because these rectangles the area that they're trying to approximate are always contained in the rectangles and these rectangles have this surplus area so they're always going to be larger than the area's that they're trying to approximate and in general if you have a function that's decreasing over the interval that we care about right over here and strictly decreasing the entire time if you use the left edge of each subdivision to approximate you're going to have an overestimate because the left edge the value of the function there is going to be higher than the value of the function at any other point in the subdivision and so that's why for a decreasing function the left Riemann sum is going to be an over s now let's think about the right Riemann sum and you might already guess that's going to be the opposite but let's visualize that so let's just go with the same three subdivisions but now let's use the right side of each of these subdivisions to define the height so for this first rectangle the height is going to be defined by G of 4 so that's right over there and then for the second one it's going to be G of 6 so that is right over there and for the third one it's going to be G of 8 and so let me shade these in to make it clear which rectangles were talking about this would be the right Riemann sum to approximate the area it's very clear here that this is going to be an under estimate under estimate because we see in each of these intervals the Riemann the right Riemann sum or the rectangle that we're using for the right Riemann sum is a subset of the area that it's trying to estimate we're not able to it doesn't capture this extra area right over here and once again that is because this is a strictly decreasing function so if you use the right endpoint of any one of these or the right side of any of these subdivisions in order to define the height that right value of G is going to be the lowest value of G in that subdivision and so it's going to be a lower height than what you could even say is the average height of the value of the function over that interval so you're going to have an underestimate in this situation now if your function was strictly increasing then these two things would be swapped around and of course there are many functions that are neither strictly increasing or decreasing and then it would depend on the function and real and sometimes even it would depend on the type of subdivisions you choose to decide whether you have an overestimate or an underestimate
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