 Welcome back to the final part of lecture 44 from math 1210 here Then we we finished section 5.1 and now we're gonna move on and start in this lecture part 5.2 Which will also be the entirety of the next lecture 45 about 5.2 here we want to talk about the definite integral We talked about previously about indefinite integrals, which is just a different name for anti derivatives What is a definite integral? Well when we have a function f in an interval a to b We define the definite integral to be the area under the function f As yes you go on the interval from a to b. So what do we mean by that? So we've seen this object before We saw this in some of the previous slides in section 5.1 here So take the sum so remember the sigma there means a sum take the sum or I goes from 1 to n of f of x i star times delta x now this f of x i star and this delta x This is the length and width of a rectangle the f of x i star is the length of the rectangle The delta x is the width of the rectangle So this f of x i star delta x is the area of the ith rectangle under the curve if you add up together all the rectangles This thing gives us an approximate area under the curve and this sum right here is called a Riemann sum Okay, and so in the previous in the previous lecture parts, especially in 5.1 We were calculating Riemann sums these areas and the curves now Those were just approximations of the area under the curve as we had these functions and we look at these rectangles, right? the rectangles do an okay job at Estimating the air into the curve, but there's always a little bit of error associated to them sometimes they overestimate sometimes underestimate Etc. Etc. Now the idea we also saw is that the more and more rectangles we use The better and better they fit under the curve the error gets smaller and smaller the skinnier the rectangles get So we want to use more and more rectangles to get better better estimates So, you know instead of using three rectangles use ten instead of ten use 17 instead of 17 use three trillion right you can keep on doing better and better than more rectangles You use well, what's the best number of rectangles to use? well That's where calculus comes into play here that we can increase the the accuracy by taking arbitrary large number of rectangles so if we were to take the limit as n goes to infinity as Each of these Riemann sums approximates there in the curve if we were to use an infinite number of rectangles We could capture the true area not an approximation But we could capture the true area under the curve So we take the limit as the number of rectangles goes to infinity and so the limit of the Riemann sum is what we define to be this Definite integral we use the same notation we use for for indefinite integrals with one small Exception here. We're still going to have the integral symbol which is this elongated s We have our function f of x. This is what we're integrating the integrand We have this differential dx, which tells us what we're trying to integrate here the variable But you'll see these numbers a and b that tell you the start and stop of the interval there So f is the function. We're trying to find the area under And that this here tells us the lower the left bound This tells us the right bound and you'll notice that this right here mimics the notation we saw in the Riemann sum Right, so we take f of x dx F f of x there is the height of the rectangle the rectangles heights are determined by the function f and the width of the rectangle Is this dx here? The idea is this dx is this infinitesimally thin number, right? It's not zero because if you take f of x times zero then The area of a rectangle be zero add a bunch of zeros together just get zero But dx is supposed to represent this really wafer thin Rectangle, right? It's so thin that we can't even measure it. It's that thin, right? It's like a piece of paper, but even thinner than that And so the area of a rectangle is going to be f of x times dx f of x is the height of the function dx This is infinitesimal number You add you get that product that's going to be a really small value But it has some positive area and then we add them together This is why the integral symbol is a long s because s stands for some Much in the same way that the sigma stands for a sum as well And so this gives us what we call a definite integral Why do we call it a definite integral the other ones the integers Indefinite integrals And we will see in the next section that definite integrals which calculate the area under the curve are related to the indefinite integrals aka antiderivatives and the previous Part of this lecture actually gives us a hint on what that connection is And i'll elaborate that a little bit more in the future And so this we say that a function is integrable If this limit exists because a definite integral is the limit of a Riemann sum limits might not exist It depends on the function f Now if this limit exists we say the function is integrable and it doesn't matter how you choose xi star In terms of approximation the xi star makes a big difference when we were approximating pi We saw that the midpoint rule was actually more accurately Proximating pi than all the other approaches we had tried and the midpoint rule does pretty good in practice But when it comes to the limit as you have infinitely many rectangles, it doesn't matter how you choose xi star So in that case we're typically going to choose xi star For the sake of simplicity, we're going to choose it to just be a plus i delta x That is we're going to be basically using r n the right endpoint rule When we do these things But in the meanwhile, what I want you to understand just just for this lecture part right now Is that the definite integral represents the true area into the curve And so let's look at some examples of that if we have the if we want to compute the integral From zero to four of the function 2x dx Well, let's take a look at that function from them geometrically. What are we trying to do? So here's our x and y axis. We want to go from zero to four Over here and our function y equals 2x That's actually the that's the that function is a linear function y equals 2x It's a it's a function which goes to the origin. That's its y intercept and its slope Would be 2 let me try that again. This is not going to be perfectly drawn to scale But hey, I didn't give a really good scale down here. So who can tell right? I don't label the y axis. No one will ever know But if this is our function y equals 2x I want you to notice that the area under the curve Of this linear function is a triangle And so we can and it's actually a right triangle. Is it not? And so we can find the area under this curve by the traditional formula That is to say this integral From zero to four of 2x dx. It is an area. It's the area of this triangle Which we could compute using the usual one half base times height formula Now the base of this triangle is going to be this distance right here It's the length of this side, which is from zero to four. So that length Is itself going to be four. So we get one half times four Well, how about the height then the height of the triangle is going to be this distance right here So how far above the x-axis do we go? Well depends on this point right here This is the point whose x-coordinate is four and whose y-coordinate will then be eight two times four And so then the height of the rectangle is going to be eight units We'll put the we'll put the base there as well. We have four units And so then we end up with a eight right here well One half of four is two and two times eight is 16 And so the area Of this triangle is going to be 16 square units That's the area under the curve and that is what this integral is calculating This integral is equal to 16 because the integral calculates the area under the curve Uh, here's another example. Let's evaluate the integral from zero to one of the function square root of one minus x squared dx And so this is trying to calculate the area under the curve. This is the area area under the curve And so it can be helpful to understand what does this curve look like And our curve is actually going to look something like the following So this right here is a function, which is it's a semi circle right To kind of see this If you take the unit circle x square plus y squared equals one If you solve for y squared you get y squared equals one minus x squared If you then solve for y you'll get plus or minus the square root of one minus x squared Choosing the plus gives you the upper semi circle and choosing the minus gives you the lower semi circle So let us make that choice right there And actually I don't even I don't need the plus then either I suppose So we're going to choose the upper semi circle, which you now see on the screen And if we want to go from zero to one X equals zero is right here x equals one is right there We're trying to calculate the area of this region right here Which this is just a quarter circle the area is going to equal one fourth Pi r squared pi r squared is there the whole circle. We're taking a quarter circle right here So we get one fourth times pi times the radius, which is one The area Under this curve is going to be pi fourths And so that is the area that's what the integral is equal to So the integral from zero to one Of the square root of one minus x squared dx. This will equal pi fourths and so this gives us some examples of integral calculations where the function coincides to a nice geometric region, which we already have a formula for Stay tuned for the next lecture Lecture 45 where we're going to talk about what do you do when the curve doesn't have a cute little Geometric formula that we've seen in the previous class. How do you find the area of the curve there? And so As always, please subscribe if you want to see more updates or you want to see more content like this one right here Leave a comment below if you have any questions like this video and I will see you next time. Have a great day everyone Keep on calculating