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Current time:0:00Total duration:8:31

AP.CALC:

LIM‑5 (EU)

, LIM‑5.B (LO)

, LIM‑5.B.2 (EK)

in the last video we attempted to approximate the area under curve by constructing four rectangles of equal width and using the using the left boundary of each rectangle the function evaluated at the left boundary to determine the height and we came up with an approximation what I want to do in this video is generalize things a bit using the exact same method but doing it for an arbitrary function with arbitrary boundaries and an arbitrary number of rectangles so let's do it so let's let me be able to draw it the diagram as large as I can to make things clear to make things as clear as possible so that's my y-axis and this right over here is my is my x-axis let me draw an arbitrary function so let's say my function looks something like that so that is y is equal to f of X and let me define my boundaries so let's say this right over here is x equals a this is x equals a and this right over here is X this is x equals B so this is B and I'm going to use n rectangles n rectangles and I'm going to use the function evaluated the left boundary of the rectangle to determine its height so for example this will be rectangle 1 I'm going to evaluate what F of a is I'm going to evaluate what F of a is so this right over here is F of a F of a and then I'm going to use that as the height of my first rectangle the height of my first rectangle so just like that so rectangle number 1 looks like this and I'll even number it rectangle 1 looks just like that and just to have a convention here because I'm going to want to label each of the boundaries of the left bound of each of the each of the x-values at the left boundary so we'll say a is equal to X naught a is equal to X naught so we could also call this point right over here X naught that that x value and then we go to the next rectangle and we could call this one right over here this x-value we'll call it X 1 it's the left boundary of the next rectangle if we evaluate F of x1 we get this value right over here this right over here is f of x1 f of x1 so it tells us our height and we want an equal width to the previous one we'll think about what the width is going to be in a second so this right over here is our second rectangle this right over here is our second rectangle that we're going to use to approximate the area under the curve that's rectangle number two let's do rectangle number three well recognize rectangle number three the left boundary we're just going to call it X sub two and its height or is going to be f of X sub two f of X sub two and its width is going to be the same width as the other ones I'm just I'm just eyeballing it right over here so this is rectangle number three this is rectangle number three and we're going to continue this process all the way all the way until we get to rectangle number n so this is the enth rectangle this is the nth rectangle right over here the nth rectangle and what am I going to label this point right over here well we already see a pattern the left boundary of the first rectangle is X sub zero the left boundary of the second rectangle is X sub one the left boundary of the third rectangle is X sub two so the left boundary of the nth rectangle is going to be X sub n minus one whatever the the rectangle number is the left boundary is X sub that number minus one and this is just based on the convention that we've defined now the next thing that we need to do in actually in order to actually calculate this area is think about what is the width so let's call the width let's call the width of any of these rectangles and for the for these purposes or the purpose of this example I'm going to assume that it's constant although you can do the sums where you actually vary the width of the rectangle but then it gets a little bit fancier so I want it equal width so I want Delta X to be equal width and to think about what that has to be we just have to think what's the total width that we're covering well the total distance here is going to be B minus a B minus a and we're just to divide by the number of rectangles that we want the number of sections that we want so we want to divide by n so if we assume this is true and then we assume that a is equal to X naught and then X 1 is equal to X naught plus Delta X X 2 is equal to X 1 plus Delta X and we go all the way that all the way to X n is equal to X n minus 1 plus Delta X then we've essentially set up this diagram right over here B is actually going to be equal to B is going to be equal to X n so this is X n it's equal to X n minus 1 plus Delta X so now I think we've set up all of the all of the notation and all the conventions that actually in order to actually calculate the area or our approximation of the area so our approximation approximate area is going to be equal to what well it's going to be the area of the first rectangle so let me write this down so it's going to be rectangle 1 so the area of rectangle 1 so rectangle 1 plus the area of rectangle 2 plus the area of rectangle 2 plus the area of rectangle 3 I think you get the point here area of rectangle 3 all the way all the way plus all the way to the area of rectangle N and so what are these going to be a rectangle 1 is going to be its height which is f of X naught or F of a either way X naught an A or the same thing so it's F of a times our Delta X times or with our height times our width so times Delta actually I can rewrite as f of X naught I wanted to write f of X naught times Delta X what is our height of rectangle 2 it's f of X 1 times Delta X f of X 1 times Delta X what's our area of rectangle 3 it's f of X 2 times Delta X and then we go all the way to our area we're taking all the sums all we had a rectangle n what's its area its f of X sub n minus 1 actually that's a different shade of orange use that same shade it is f of X sub n minus 1 times delta X and we're done we've written it in a very general way but to really make us comfortable with the various forms of notation especially the types of notation you might see when people are talking about approximating areas or sums in general I'm going to use the traditional Sigma notation so another way we could write this as the sum this is equal to the sum the sum from and remember this is just based on the conventions that I set up I'll let I count which rectangle were in from I equals 1 to n and then we're going to look at each rectangle so the the first rectangle that's rectangle 1 so it's going to be f of it's going to be f of well if we're in the eighth rectangle then we're going to take the left boundary is going to be X sub I minus 1 times Delta times Delta X and so here right over here is a general way of considering of thinking about approximating the area under a curve using rectangles where the height of the rectangles are defined by the left boundary and this tells us it's the left boundary and we see for each if this is if this is the ice if this is the ice rectangle right over here if this is rectangle I then this right over here is X X sub I minus 1 and this height right over here is f of X X f of X sub I minus 1 so that's all we did right over there times Delta X and then you sum all of these from the first rectangle all the way to the end so hopefully that makes you a little bit more comfortable with this notation we're not doing anything different than we did in this first video which was hopefully fairly straightforward for you we have just generalized it using a little bit more mathy notation