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# Worked example: 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

the continuous function G is graphed we're interested in the area under the curve between x equals negative 7 and x equals 7 and we're considering using Riemann sums to approximate it so this is the area that we're thinking about in this light blue color order the areas from least on top to greatest on bottom so this is a screenshot from Khan Academy exercise where you would be expected to actually click and drag these around but it's just a screenshot so what I'm gonna do instead of dragging them around I'm just gonna write numbers ordering them from least to greatest where one would be the least and then 3 would be the greatest so pause this video and try to think about these which of these is the least which is in the middle and which is the greatest so let's just draw out what a left Riemann sum a right Riemann sum would actually look like and compare it to the actual area and we could do an arbitrary number of subdivisions I would encourage us to do a few because you were because we're just trying to get a general sense of things and they don't even have to be equal subdivisions so let's start with a left Riemann sum so we want to start at x equals negative 7 and we want to go to x equals 7 well let's say that this is the first rectangle right over here so this is our first subdivision and it's the left Riemann sum so we would use the value of the function at the left end of that subdivision which is negative 7 x equals negative 7 the value of the function there is 12 and so this would be our first rectangle you already get a sense that this is going to be an overestimate relative to the actual area and so the next subdivision would start here so this would be our height of our rectangle and let's say they don't have to be equal subdivisions they often are but I'm going to show you unequal subdivisions just to show you that this is still a valid Riemann sum and once again this is an over estimate where the actual area that we're trying to approximate is smaller than the area of this rectangle and then let's say this third subdivision right over here starts right over there at x equals 3 and we use the left end of the subdivision the value of the function there to define the height of the rectangle and once again you see it is an over estimate so the left Riemann sum is clearly an over an over estimate and it's pretty clear why this function is this function never increases it's either decreasing or it looks like it stays flat at certain points and so for a function like that the left edge the value of the function at the left edge is going to be just as high or higher than any other value of than any other value the function takes on over that interval for the subdivision and so you get left with all of this extra area that is part of the over estimate or this area that is larger than the actual area of that you're trying to approximate now think about a right Riemann sum and I'll do different subdivisions let's say the first subdivision goes from negative 7 to negative 5 and here we would use the right edge to define the height so f of negative 5 or G of negative 5 I should say so that's right over there that's our first rectangle maybe our next rectangle the right edge is zero so this would be it right over there and then maybe maybe you'll do four rectangles maybe our third subdivision the right edge is that X is equal to three so it would be right over there and then our fourth subdivision let's just do it at x equals seven and we're using the right edge of the subdivisions remember this is a right Riemann sum so we use the right edge the value of the function there is just like that and now you can see for any one of these subdivisions our rectangles are under estimates of the area under the curve under under estimate and that's because once again in this particular case the function never increases it's either decreasing or staying flat so if you use the value of the function at the right edge it's going to be smaller it's going to be it's never going to be larger than the value that the function takes on in the rest of that subdivision and so we are continuously under estimating we're missing this all of all of this area right over there is not being included so we have an under estimate so if we want to rank these from least to greatest well the right Riemann sum is the least it is under estimating it then you have the actual area of the curve which is just the area of the curve and then you have the left Riemann sum which is the over estimate
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