 Welcome back to our lecture series Math 1220, characters two for students at Southern Utah University. As usual, I will be your professor today, Dr. Andrew Missildine. In lecture 19, we'll be talking about the idea of improper integrals. This can be found in section 7.8 of Jane Stewart's calculus textbook here. What does it mean for an integral to be improper? Imagine we have some continuous function f, and we're interested in integrals of the following form. Let's find the area under the curve from some fixed finite value a, but we go off towards infinity, right? Visually, we're thinking of something like the following. We have our function. He does something like this, and maybe because the function's asymptotic to the x-axis, we might be interested, well, if we allow this region to go off towards infinity, could it be possible that this is a finite area? Well, it turns out with the right function, it can be, and sometimes it's not, right? What we mean here is we take the integral from a to infinity. Really, this is a limit calculation. Whenever you're working with infinity, really, we're taking the limit of some kind. We're taking the limit as the upper bound of the integral goes off towards infinity. That's what a to the infinity means for an integral. Similarly, we can define the integral from negative infinity to b, right? Again, this really is just shorthand for a limit. Take the limit as the lower bound a goes towards negative infinity, and we compute that right there. In some situations, we actually want a double infinity, right? We could have a situation of a graph, maybe it looks something like the following. It's asymptotic on the right to the x-axis. It's asymptotic on the left. This is like your standard bell curve we see in statistics and probability. Could the area under the entire function be finite? Well, in the case of the normal distribution, the area under the curve is equal to one. In this situation, if you're going from negative infinity to infinity, what you do is you just break this up into two integrals themselves. You go from negative infinity to c, which is defined by this rule, and then you also take the integral from c to infinity, which is defined by this rule right here. Typically, you take c to be zero because that's a good simple choice, but it could be any number you want to. These are examples of improper integrals. Now, as improper integrals are defined using limits, if this limit exists, we say that the integral is convergent. Now, if this limit does not exist, which also includes like maybe the limit does not exist, or if the limit itself is infinity, we'd still say the limit doesn't exist because infinity is not a real number, then we say that the improper integral is divergent. So you'll hear this language, the function or the integral is convergent or divergent. Now, this is describing the integral, not the function themselves. The function might converge towards the x-axis like you see in this picture right here, but the area of the curve could still be infinite and therefore the integral is divergent. Let's see some examples of such things, right? So let's take the integral from one to infinity of x to the negative three halves over, the negative three halves power. Now, by definition, an improper integral is the limit as b approaches infinity of the integral one to b, x and negative three halves here. The reason why we put so much emphasis on the limit here is that the fundamental theorem of calculus does not apply in this setting where you're going off towards infinity. The fundamental theorem of calculus only applies to proper integrals for which this is right here, now a proper integral because b is a finite number, it gets bigger, bigger, bigger, but it's a finite number that the fundamental theorem of calculus applies here. Now, this is a subtlety that's sometimes lost on beginners to calculus, but it is an important distinction here. We can apply the fundamental theorem of calculus in this setting, right? For which as an anti-derivative, we're gonna raise the power of x by one, so it becomes negative one half. We divide by negative one half. Well, of course, this is the same thing as times by negative two, and we're going to evaluate from one to b. So we end up with the limit as x approaches, or sorry, as b approaches infinity of negative two x to the negative one half power. And I'm actually gonna write this as negative two over the square root of x as you go from one to b here. And so when you plug these, notice there's a negative sign, so I'm gonna switch the order. So we get the limit as b goes to infinity of two over the square root of x as you go from b to one. And so then this will look like, because when you plug in the one, you're gonna get two over the square root of one. You don't need a limit for that. You're gonna track the limit as b goes to infinity of two over the square root of b right here. And so now calculating that limit there, square root of one is one, two over one is two, so you get two minus. Well, you're gonna get something that looks like two over the square root of infinity. Now some people will get really scared right now because doing arithmetic with infinity is sort of the mathematical equivalent of doing black magic, right? You know, if we had a chemistry professor that was teaching us alchemy in our chemistry labs, we'd be really concerned about that. Well, this is sort of like a dark numerology that we're doing right now at arithmetic with infinity. But with that, we still get a little bit of a comfort with this. When you divide by infinity, that's gonna give you zero. So two minus zero gives you two. The improper integral is gonna add up, this improper integral is two. The area under this curve as you go from one to infinity is two. It's quite fascinating that even though the length of the interval is infinite, the area under the curve is still a finite value. And we get this two right here. Now, some things I do wanna mention about this is that properly, if you're gonna be a proper improper integral, you should be using this limit notation. But frankly speaking, if most people are gonna feel more comfortable with the following, we're gonna do an improper improper integral. We sort of recognize that, oh, when there's an infinity, that means we take a limit, even if we don't write that. Looks like what do we define this notation to mean? It means take a limit, okay? So many of us are just gonna jump straight to the anti-derivative. We get negative two x to the negative one half as you go from one to infinity. Some people have a spaz attack right now when they see this is like, what are you doing with infinity, right? What does that mean? Well, it means a limit. It means a limit, just like it did here. Why can't we do it there as well? In which case, then the next step looks like negative two x, sorry, negative infinity to the negative one half plus two times one to the negative one half right there. And so again, that people are still panicking like, oh no, what's going on? You can't do arithmetic with infinity. Well, we kind of did already. We were used to doing that. And yes, what it means is this right here. So if we wanna be completely proper, we're taking limits and this is just supposed to be an abbreviation of a limit. So please chillax, take a chill pill, right? In which case then you end up with zero plus two which gives you two as well. So as we go through these calculations, I'm gonna try this more simplified notation but recognize like in this one, you integrate from negative infinity to negative two here. This means we're taking the limit as a approaches negative infinity here of the integral a to negative two one over x squared dx. That's what it means, but that full blown out notation isn't really gonna make much of a difference for us. We can, we're gonna prefer to use the abbreviation that I mentioned earlier. So if we integrate from negative infinity to negative two, well, let's look for an anti-derivative. I'm gonna prefer to write this as a power x and negative two to make it a little bit easier. In which case then the anti-derivative will look like negative x and negative one as you go from negative infinity to negative two. Or if you prefer, we can write this as negative one over x as your anti-derivative as you go from negative infinity to negative two. Like so plugging in the numbers, you're gonna get negative one over negative two plus one over negative infinity. Like we saw before, if you divide by infinity, I cannot draw infinity right now. When you divide by infinity, this is gonna go to zero. And so we're left with what looks like, which looks like a one half, right? So we get one half as our final result right here. As a third example, what happens if you wanna go from negative infinity to infinity? In this situation, it really is important that you break this thing up. Go from negative infinity to zero, one over one plus x squared dx plus the integral from zero to infinity, one over one plus x squared, like so. You do need to break it up because the thing is if you're too careless with the limit notation here, you can actually get something that looks convergent that's truly divergent. So one has to be very careful about this. Now it is the same function in both situations, one over one plus x squared. The anti-derivative is gonna be arc tangent, arc tangent of x, as you go from negative infinity to zero, and you add to that arc tangent as you go from zero to infinity. Arc tangent of zero is itself zero as arc tangent of infinity. What does one mean by that? Well, we're really looking for the horizontal asymptote as x approaches infinity on the right side of the graph. And then you're going to subtract from this arc tangent of negative infinity, like so. And the horizontal asymptote for arc tangent is gonna be a pi over two. And then you're gonna, on the other side it's gonna be a negative pi over two. And so this adds up to be pi, which is kind of fun, right? The area under the curve here is pi. Now one thing I noticed is that as I proposed the slides, there's actually an example I skipped by mistake. I don't know how I did that. I didn't prep it beforehand, but I do wanna, this is an important example to see here. Let's do, this will be example D. Let's do the integral from one to infinity of dx over x. So the important thing to recognize here is we wanna integrate from one to infinity. It's an improper integral, right? So we have one over x dx. The anti-derivative one over x is the natural log. As you go one to infinity. But what happens this time as we plug in infinity? You get the natural log of infinity minus the natural log of one, which the natural log of one is itself zero. So that'll disappear. So the area under this curve is whatever the natural log infinity means, which as we mentioned before, the natural log infinity really means we're taking the limit as x approaches infinity of the natural log of x. So what happens to the natural log as x gets bigger, bigger, bigger, bigger, bigger? This thing is gonna go off towards infinity. So therefore this is an example of a divergent, a divergent improper integral. The other three examples that we saw were examples of convergent integrals because we ended up with a positive number when we were done. Divergence is a possibility and do be aware that if your integral turns out to be infinity, these improper integrals often turn out to be infinity. It is an example of a divergent integral because it doesn't converge towards a number. The area under the curve is in fact finite.