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Current time:0:00Total duration:5:21

AP.CALC:

FUN‑6 (EU)

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we've already seen and you're probably getting tired of me pointing it out repeatedly that this yellow area right over here this area under the curve Y is equal to f of X and above the positive x-axis or I'm just going to say just above the x-axis between between the point between x equals a and x equals B that we can denote this area right over here is a definite integral from A to B of f of X DX now what I want to explore in this video and it'll come up with kind of a answer that you probably could have guessed on your own but at least get an intuition for it is I want to start thinking about the area under the curve that's a scaled version of f of X let's say it's Y is equal to C times f of X Y is equal to some some number times f of X so it's scaling f of X and so I want this to be kind of some arbitrary number but just to help me visualize you have to draw something so I'm just going to kind of in my head let's just pretend like the C is 3 just for visualization purposes so it's going to be 3 times it so instead of 1 instead of this far right if it's going to be about this far far right over here instead of this far right over here it's going to be that another right over there and then instead of it's going to be about there and then instead of it being like that it's going to be 1 2 and then 3 right around there so I'm trying to get a sense of what this curve is going to look like it's going to a scaled version of f of X and for at least what I'm drawing is pretty close to 3 times f of X but just to give you an idea it's going to look something like and over here let's see if this distance do a second one a third one's going to be up here it's going to look something something like this it's going to look something like that so this is a scaled version and the scale I did right here I assumed to positive I assumed the positive C greater than 0 but this is just for visualization purposes now what do we think the area under this curve is going to be between a and B so what do we think this what do we think what do we think this area this area right over here is going to be and we already know well how we can denote it that area right over there is equal to the definite integral from A to B of the function we're integrating is C f of X DX I guess to make the question a little bit a little bit clearer how does this relate to this how does this green area relate to this yellow area well one way to think about it is is we just scaled the vertical dimension up by C so one way that you could reason it is well if i if i'm finding the area of something if i have the area of a rectangle and i have the vertical dimension is let's say the vertical dimension is i don't want to use those same letters over and over again well I'll just well let's say the vertical dimension is alpha and the horizontal dimension is beta we know that the area we know that the area is going to be alpha times beta now if I scale up the vertical dimension by C so instead of alpha this is C times alpha and this is the width is beta if I scale up the vertical dimension by C so this is essentially this is now C times alpha what's the area going to be well it's going to be C alpha times beta or another way to think of it I have just taken when I scale one of the dimensions by C I take my old area and I scale up my old area by C and that's what we're doing we're scaling up the vertical dimension by C when you multiply C times f of X f of X is giving us the vertical height now obviously that changes is our X changes but when you think back to the Riemann sums the f of X is what gave us the height of our rectangles we're now scaling up the height or scaling I should say because that we might be scaling down depending on the C we're scaling it we're scaling one dimension by C if you scale one dimension by C you're going to scale the area by C so this right over here the integral let me just rewrite it the integral from A to B of C f of X DX that's just going to be the scaled we're just going to take the area of f of X so let me do that in that same color we're going to take the area under the of f of X from A to B f of X DX and we're just going to scale it up we're going to scale it up by this C we're just going to scale it up by the C so you might say okay maybe I kind of felt that that was you know if it's a C inside the integral now I can take the C out of the integral once again this is not a this is not a rigorous proof based on the definition of the definite integral but it hopefully gives you a little bit of intuition why you can do this if you scale up the function you're essentially scaling up the vertical dimension so the area under this is going to just be a scaled up version of the area under the original function f of X and once again really really really useful property really really useful property of definite integrals that's going to help us solve a bunch of definite integrals and and and and and kind of clarify what we're even doing with them