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# The fundamental theorem of calculus and definite integrals

AP.CALC:
FUN‑6 (EU)
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FUN‑6.B (LO)
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FUN‑6.B.1 (EK)
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FUN‑6.B.2 (EK)
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FUN‑6.B.3 (EK)

## Video transcript

let's say we've got some function f that is continuous over the interval between C and D and the reason why I'm using C and D instead of a and B is so I can use a and B for later and let's say we set up some function capital f of X which is defined as the area under the curve between C and some value X where X is in this interval where where F is continuous under the curve so it's the area under the curve between C and X so if this is X right over here under the curve f of T DT so this right over here f of X is that area that right over there is what f of X is now the fundamental theorem of calculus tells us that if F is continuous over this interval then f of X is differentiable at every X in the interval and the derivative the derivative of capital f of X and let me say that let me be clear capital f of X is differentiable at every possible X between C and D and the derivative of capital f of X is going to be equal to lowercase F of X fair enough now what I want to do in this video is connect the first fundamental theorem of calculus to the second part or the second fundamental theorem of calculus which we tend to use to actually evaluate definite integrals so let's think about let's think about what F of f of B what F of B minus F of a is what this is where both B and a are also in this interval so f of B and we're going to assume that B is larger than a so let's say that B is this right over here we've done the same color so let's say that B is right over here F of B is going to be equal to we just literally replace the B where you see the X it's going to be equal to the definite integral between C and of F of T DT which is just another way of saying it's just another way of saying is the area under the curve between C and B so this F of B capital F of B is all of this business all of this business right over here and from that we are going to want to subtract we are going to subtract capital F of a which is just the integral between C and lowercase a of lowercase F of T DT so let's say that this is let's say that this is a right over here capital F of a is just literally the area between C and a under the curve lowercase F of T so it's this right over here it's all of this business all of this right over here so what is if you have this blue area which is all of this and you subtract out this red this this this this magenta area what are you left with well you're left with this green area you're left with this green area right over here and how would we represent that how would we denote that well we could denote that as the definite integral between a and B of F of T of F of T DT and there you have it this right over here is the second fundamental theorem of calculus it tells us that this if F is continuous on the interval that this is going to be equal to the antiderivative or an antiderivative of F and we see right over here that capital F is the antiderivative of F so that we could view this as capital F anti antiderivative just is this is how we define capital F the antiderivative or we didn't define it that way but the fundamental theorem of calculus tells us that capital F is an antiderivative of lowercase F so right over here this tells you if you have a definite integral like this it it completely equivalent to an antiderivative of it evaluated at F and from that evaluated at B and from that you subtract it evaluated at a so normally it looks like this I just switched the order the definite integral from A to B of F of T D T is equal to an antiderivative of F so capital F evaluated B and from that the antiderivative from that subtract the antiderivative evaluated at a and this is the second part of the fundamental theorem of calculus or the second fundamental theorem of calculus and it's really the core of an integral calculus class because it's how you actually evaluate definite integrals