Definite integral evaluation
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The fundamental theorem of calculus and definite integrals
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 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 of capital F of x-- and 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 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. And we'll do that in 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 b of f of t dt, which is just another way of saying the area under the curve between c and b. So this F of b, capital F of b, is all of this business right over here. And from that, we are going to want 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 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 right over here. So if you have this blue area, which is all of this, and you subtract out to this magenta area, what are you left with? Well, 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 dt. And there you have it. This right over here is the second fundamental theorem of calculus. It tells us that 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 we could view this as capital F antiderivative-- this is how we defined 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's completely equivalent to an antiderivative of it evaluated at b, and from that, you subtract it evaluated at a. So normally it looks like this. I've just switched the order. The definite integral from a to b of f of t dt is equal to an antiderivative of f, so capital F, evaluated at b, and 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.