 In this part of lecture 45, let's talk a little bit about properties of definite integrals and some of these properties you're going to have seen before. So some of them are going to look pretty familiar, but let's talk about them for a second. So some properties of definite integral the first one. Let's consider the integral of the function f of x dx. And what happens if we go from a to a that is the starting and stopping number of the of the domain is the same number. So think about that for a second. We have some function f right here. We have our x axis made that somewhere below. And we take this number a and a, right? We're looking for the area under the curve from a to a. Well, that's just going to be a sliver. It's just a line segment. What's the area of a line segment? Well, it's nothing, right? There is no area to align segment. And hence you get an area of zero. So if you integrate from a number to itself, the area under the curve is always going to be zero independent of whatever the function is. All right. Now, here's another one. This one's kind of interesting that we actually use this one all the time surprisingly. If you take the integral from a to b of f of x dx, this is equal to negative the integral from b to a f of x dx. So what you're going to notice here is that if you take the limits of the integral a and b and you swap them around, you actually get negative the original integral and negates everything. And why is that? Well, this number right here, you know, this dx often goes without notice. Sometimes calculus forget about dx here. But remember, where does this, where does this integral come from? The integral is the limit as n goes to infinity of the Riemann sum n equals 1 to infinity, sorry, 1 to n of f of x i delta x. So the idea here is I want you to see the similarity between the integral notation and this Riemann sum. You have this big sigma here. What does sigma mean? Sigma means sum. Sigma is the Greek letter for s. So s is for sum. When you look at the integral right here, what does this integral symbol look like? It looks like a big s. What is s for? s is for sum. And so we use that notation the same. And then what follows the sum? You have the length of your rectangle times by the width of your rectangle. Length times width, this f of x i times delta x is length times width. What about this f of x dx? This is still just length times width. The notation is almost the same thing. The only thing that's missing is that this limit as the number of rectangles goes to infinity, this is concealed inside of this definite integral notation. That's why we use the long s as opposed to the sigma to represent the limit being taken. But even still, it always breaks down its length and width. What is our delta x after all? Delta x by definition was b minus a over n. Well, that's where b was on the top right here and a was on the bottom. Well, if we switch the roles, if we put a on top and b on bottom, you're going to get a minus b over n, which if you factor out a negative sign, that will switch the order back to where it was b minus a over n. This gives you a negative delta x. And so as you take the limit as n goes to infinity, delta x is going to converge towards dx here. And so that's the idea here is if you switch the orders, the delta x gets swapped around and then the subsequent dx will also gain this negative sign as well. So if you integrate backwards, you will get negative the area of the original region. And like I said, that's a trick that we actually do use on occasion here. These next two properties you hopefully might recognize if not such a big deal here, but look at property three and property four. This tells us property three that if there's a scalar inside of the integral, you can factor it out and you get a scalar on the outside. This one right here says if you add together two functions inside the integral, this is the same as adding together the integral separately or taking the difference of those. If we put these two principles together, this one right here and this one right here, these two powers come together to give us linearity. That is the definite integral is a linear operation. So as the indefinite integral anti derivatives, sigma limits derivatives, there's so many linear operations in in calculus, right? And so people who want to kind of explore this much deeper, I actually recommend you take a course on linear algebra, which explores this idea of linearity that seems to pop up over and over and over and over again in calculus. Calculus is sort of based upon the premise that we can solve linear problems and we can approximate nonlinear problems using linear problems. This idea of area under the curve is exactly the same idea. We have some type of nonlinear curve, but we can approximate it using these linear approximations which are rectangles, these polygons, right? We can approximate the area under the curve using a linear approximation take limits to get the correct value. So linear answer is actually really important to calculus students. I recommend you look into it in the future. This last property I want to mention, this one also somewhat intuitive here. If you take the integral from a to b of f of x dx, this is the same thing as the integral of a to c of f of x dx plus c to b f of x dx. So providing a picture of what's going on here, here's our function f. Here is our value, our domain a to b. And so we want to find the area under this curve from a to b right here. Well, this idea just says the following, if we're going to cut this into two pieces at some intermediate value c, then the area from a to c plus the area from c to b gives you the area from a to b. You can add those things together. But I also want to mention that this also applies that even if c is not between a and b, that's perfectly fine. Because of this property right here, if you integrate backwards, you'll, you'll double count this area, you'll count it once and then subtract it. So the cancel in the end will be the same. And so I want to show you an example how we can use these principles and our integral calculations. So let's calculate the integral from zero to one of four plus three x squared dx. Well, by properties of integration, we can break this up into two integrals. So you integrate from zero to one of four dx plus the integral from zero to one of three x squared dx. That is integral of a sum is a sum of integrals. And then this constant coefficient that sits out in front can come out as well. And we can do that here as well. We can bring these things out. And so then you end up with something like four times the integral from zero to one of dx plus three times the integral from zero to one of x squared dx right here. And so because of linearity, we can break this up into smaller pieces like so. And so how do we compute these individual integrals? Well, one thing to notice is that in a previous video, we had exactly done this one, the integral from one, from zero to one of x squared. And we found that that was a third. And there was a Riemann sum calculation there, but we're going to just use that right here and that this integral is equal to a third. How about this one right here? Right, you don't see a function. That's just because the function here is one. Geometrically, what we're saying is the following, we have the horizontal line y equals one. And then underneath it, the x-axis, we want to go from zero to one. And so we're considering this region right here. We're trying to find this area, but this right here is just a rectangle. So if we're trying to find the area of a rectangle, we're just going to take length times width. The length here is a one. The width right here is likewise one. So actually, this isn't just a rectangle. This is just square, right? And so we see that this integral right here, the integral from zero to one of one is just the number one itself. But four times one plus three, sorry about that, three times a third. And so simplifying this, we get four plus one and we see that the area under the curve is going to equal five. So we can use linearity to help us compute integrals, definite integrals, much like we did anti-derivatives, derivatives, and the like. So feel free to use these properties in your calculations.