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# Integrating sums of functions

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.A (LO)
,
FUN‑6.A.1 (EK)
,
FUN‑6.A.2 (EK)

## Video transcript

so we have a couple of functions here this is the graph of y is equal to f of X this is the graph of y is equal to G of X and we already know something or we know we know ways to represent the area under the curve y equals f of X between these two points X is equal to a and X equal to B so this area right over here this area right over here between the curve and the x-axis between x equals a and x equals B we know we can write that as the definite integral from A to B of f of X DX and we can do the same thing over here we could call we could call this area let me pick a color that I have not used well this is slightly different green I could call this area right over here the area under the curve Y is equal to G of X and above the positive x-axis between x equals a and x equals B we could call that the definite integral from A to B of G of X DX now given these two things let's actually think about the area under the curve of the function created by the sum of these two functions so what do I mean by that so let me this is actually a fun thing to do let me start again that's that's exactly what we have over here this is the graph of y is equal to f of X but what I want to do is I want to approximate the graph of y is equal to so my goal is to graph y is equal to f of X f of X plus G of X plus G of X plus G of X so for any given X it's going to be f of X so that's the f of X and I'm going to add the G of X to it so what would that look like so that's going to look like let's see at when x is 0 G of X looks like it's about that length right there I'm obviously approximating it so I'm going to have to add that length right over here so I'd probably be right around there at x equals a it's a little bit more and so but now my f of X curve has gone more it has has increased but if I take that same distance above it if I add the G of X there right about there once again I'm just eyeballing it trying to get an approximation give you an intuition actually for what f of X plus G of X is I'm just trying to add G of X for given X now let's see if I'm a little bit let's say that I'm between a and B G of X is about that distance right over there so if I wanted to put that same distance right over here it gets me right about it gets me right about there and then when X is equal to B G of X is about that vague so I have to add that length which is about which it looks something something like that that actually looks like a little bit too much maybe something like that so if I were to add the two if I were to add the two I get a curve that looks something like this that looks something like this and maybe it just keeps on going higher and higher so this is the curve or it's a pretty good approximation of the curve of f of X plus G of X now an interesting question is is what would be well we know how we can represent this area so the area under the curve f of X plus G of X above the positive x axis between the between x equals a and x equals B we know we can represent that as let me see I have not used pink yet so this area right over here we know that that could be represented as the definite integral from A to B of f of X of f of X plus G of X plus G of X DX now the question is how does this thing relate to D or how does this area relate to these areas right over here well the important thing to realize is this area that we have in yellow that's going to be this area right over here that one's pretty clear that one's pretty clear but how does this area in green relate to this area there and to think about that we just have to think about what does it integrate what is it will mean remember what does it represent we've already thought about these like these really small rectangles we're taking the sum of an infinitely number or the limit as we get an infinite number of these infinitely thin rectangles but when we're thinking about Riemann sums we're thinking about okay we have some change in X and then you multiply it times essentially the height which is going to be the value of the function at that point well over here you could have the same change in X you could have the exact same change in X and what is the height right over here well that's going to be this exact height right over here we saw that when we constructed it this is going to be the G of X at that value so even though the rectangles look like they're kind of shifted around a little bit and they're actually all shifted up by the f of X the heights of these rectangles that I'm drawing right over here are exactly the same thing as the heights of the rectangles that I'm drawing over here they once again they are all just shifted up and down by this by this f of X function but these are the exact same rectangles or they have the exact same heights and the limit as you get more and more of these by making them thinner and thinner is going to be the same as the limit is getting more and more of these as you get thinner and thinner and so this area right over here and I'm obviously not doing a rigorous proof I'm giving you the intuition for it is the exact same thing as this area right over here so the area under this curve the the definite integral from A to B of f of X plus G of X DX it's just going to be the sum of these two is just going to be the sum of these two definite integrals and you might say oh this is obvious well you know one way or maybe it's not so obvious but what is this actually useful well as you later actually learn to evaluate these integrals you'll see that one of the most powerful ideas is being able to decompose them in this way to say okay if I'm taking if I'm taking the definite integral from 0 to 1 of x squared plus sine of X which you may or may not have learned to do so far you can at least start to break this down you could say okay well this is going to be the same thing as the integral from 0 to 1 of x squared DX plus the integral from 0 to 1 of sine of X DX and you see this is one of the most powerful principles of definite girls to make them to it when you start to try to compute them or even sometimes conceptualize what they are representing