 Welcome back to our lecture series, Math 1210, Calculus I for students at Southern Utah University. As usual, I am your professor today, Dr. Andrew Misseldine. This lecture is actually an interesting lecture because it's not just the 47th lecture for Calculus I. This also will serve as the first lecture for Calculus II, Math 1220 for students at Southern Utah University. This is a little bit of the overlap here because the fundamental theorem of calculus, part two, or in Calculus II, we'll just call this the fundamental theorem of calculus, serves as really like the bread and butter for a lot of the type of calculations we're gonna be doing in the future. Calculations of definite integrals is something we do all the time in Calculus II. It's an important principle for Calculus I, but it's worth reviewing as we go into Calculus II. So this lecture is gonna serve both purposes in that regard. Let me slide up the slides a little bit. Let's talk about the fundamental theorem calculus, part two. We've seen the first part of the fundamental theorem of calculus which determines that derivatives and integrals are actually inverse operations. The second part shows us how to use anti-derivatives to compute definite integrals. So the fundamental theorem of calculus, part two says this specifically. If F is a continuous function, continuity is necessary for a couple of reasons. We'll see a reason why later on, but in particularly the fundamental theorem of calculus, part one uses continuity here. And the part two is gonna follow very quickly from part one. So if F is continuous on a domain A to B, then the definite integral, that is the integral from A to B of F of X DX can be computed as capital F of X, which you see this vertical line here, AB there. That's just gonna be a shorthand for this right here. Capital F of X, you're gonna take F of B minus F of A. So again, kind of pointing out here that this vertical line notation you see is just gonna be shorthand for plug in the numbers A and B and subtract them. Where this capital F of X is an anti-derivative of little F, that is capital F prime equals little F prime. So what the fundamental theorem calculus tells us right here is that if we wanna compute a definite integral, what we can do is we can find any anti-derivative, any anti-derivative here. And with any anti-derivative, we can evaluate at the end points of the interval A and B and take the difference. And this will help us in terms of our calculation of the definite integral. And so I wanna talk about the proof of this statement a little bit. And so the exact proof you can see to the left side of the screen, I'm gonna kind of summarize it for us here, you're welcome to read through this or you can also download the script of this lecture which is you can find in the comments below. And so let's take some area function G of X equals the integral from A to X of F of T DT. These type of integral functions were the main focus of the fundamental theorem calculus part one. And so that's what we're gonna use right here. We see that if we take the derivative of G, this will just equal little F of X. And that follows by FTC one. And so what this tells us is that this function G of X, this is an anti-derivative of the function F. It's an anti-derivative. And it's not, I mean, because there's not one unique anti-derivative, we've seen that plus C before. This constant A kind of keeps track of if we had different A's, we would get different anti-derivatives. So we just have an anti-derivative right here. And so we know that all anti-derivatives only differ by a constant, right? So if you have two anti-derivatives, capital F and capital G are something like that. Or in this case, lowercase G is our anti-derivative. It's anti-derivative. Anti-derivatives only differ by a constant. So capital F of X will equal little G of X plus a constant. That's how they're only gonna differ by that constant right here. And so we know that any anti-derivative of little F will look like this. It'll look like this G of X plus a constant. And that's not gonna make any big difference in terms of the calculations, the plus constant right here. And we'll talk about that in just a second. So using the continuity right here, using the continuity of these anti-derivatives, we're also gonna get that F of A is gonna equal G of A. Oops, G of A plus a constant. And we can do this for B as well. So we can evaluate the function at these locations and get these things right here. All right, so I want you to look at this calculation right here and we're gonna focus over here because this one explains it well enough. If you take the difference F of B minus F of A, well by the identities we identified over here, F of B is the same thing as G of B plus C. And F of A is just the same thing as G of A plus C. And so when you take the difference of these things, notice that you're gonna get a plus C and a minus C. Those are gonna cancel out giving us this statement right here. This is why it doesn't matter which anti-derivative you get here. When it comes to the funnel theorem of calculus, you can use any anti-derivative because the plus C would cancel out. That doesn't mean we should never ever consider the plus C. If we're talking about an indefinite integral, the general anti-derivative family, you do need that plus C. But for definite integrals, any anti-derivative works here and you could assume C is zero in such a situation. And so then if you look at the definition of G, right? G of X remember is this function that we saw earlier, it's the function where you take the integral from A to X of F of TDD, right? And so we're gonna plug in for this X the value A and B what you see here and here. Now, if you evaluate an integral from A to A, it doesn't matter what the function is, that thing's gonna equal zero, right? So essentially this is gone. And so you're left with the original expression F of B minus F of A is equal to the integral from A to B of F of TDD. And so typically we're gonna work this way. We start with the integral. If we can compute an anti-derivative, we can evaluate the limits of the integral at the anti-derivative and taking their difference that'll give us the definite integral. It'll give us the area under the curve. And so there's two important takeaways I want you to get from this fundamental theorem calculus with its proof. The first is I've already mentioned is that, well, let me, I haven't mentioned this, but let me say some more about this. Oh, I mean, no, we talked about this. And what we'll say it again here is that with the fundamental theorem, you can use any anti-derivative you want of little F. It doesn't matter which one you use. And so suppose that you have an anti-derivative of F like we saw a moment ago when you take this F of X plus C, the plus C's are gonna cancel out. And so you just end up with F of B minus F of A. The plus C you're allowed to ignore when it comes to definite integrals. So with definite integrals, you don't need C. No plus C is necessary, but for indefinite integrals, for these indefinite integrals, let me emphasize you must have the plus C. Cause the indefinite integral, this is looking, this is the general anti-derivative of the function. And so you'll often look at this as the integral of F of X DX. You'll notice that the numbers here are missing. You don't see any there whatsoever. The definite integral on the hand, you will see the bounds, the integral from A to B of F of X DX. It's important that these things are related as the fundamental theorem tells us, but they're not exactly the same thing. For example, the definite integral is gonna give us a number because we're looking for the area under a curve. The indefinite integral, on the other hand, gives us a family of functions. And those functions are gonna be the anti-derivatives of little F. And that's why the plus C is necessary because without the plus C, you're not getting the general family. So that's one remark I wanna mention. A second remark to mention here is that the fundamental theorem of calculus does not give us the definition of the integral. The definite integral, like we said, is the area under the curve. It's a number, it's a limit of a Riemann sum. That's the definition, that limit of Riemann sums. The fundamental theorem of calculus just gives us a tool to calculate the area under the curve using anti-derivatives. It doesn't say that definite integrals are equal. It doesn't say that they are anti-derivatives. Definite integrals is limits of Riemann sums. And that distinction will be very helpful as we work with definite integrals and indefinite integrals in the future.