 Hey everyone, I want to talk some more about approximate area of the curve, but I want to offer a different one more way of trying to calculate the area of the curve becomes again to this Xi star. So imagine we have this function f of x equals one plus x squared, and we go from the interval negative one to one, and we have four intervals right there for some intervals. So if we were to graph that thing. Here's my x and y axis. The function f of x equals one plus x squared. That's just your standard parabola, but it was shifted up by a factor of by by one unit up the y axis. You might get something like this. So this is our function f. And so we want to go from negative one to one. We're trying to estimate this area right here. It should look more symmetrical than when I've drawn right here and that's okay. So we talked about like the left rule, the right rule, the midpoint rule and the trapezoid rule before, but there's one other way I want to kind of talk about this thing. If we were to break this up into four intervals from going from negative one, negative one to one here, we would take for our delta x one minus negative one divided by four. This ends up with two over four, which is one half. This is our delta x value. So we basically going to cut this into force like this. And so we have these intervals that cut everything into pieces like this. That is a crude drawing here. And so one way we can determine the height of these rectangles is by using the left rule, the right rule, the midpoint rule, all these type of things. But what we're going to talk about here, the so-called lower sum and upper sum, is that we're going to use our sampling points x i star to be those that optimize the interval. So for the lower sum, we'll choose x i star so that f x i star is always minimum. We can minimize that. So for this picture, if you look at the first interval, we would choose this value right here, which would be the minimum of the first interval. Then you choose this value to be the minimum of the second interval. You choose this value again to be the minimum of the third interval, and then you'll choose the minimum of the fourth interval to be that point right there. So you're always choosing this minimal thing here. So for this lower sum, you end up with one half times f of, you get negative point five plus f of zero plus f of zero. It actually shows up twice because it's the minimum twice, and then you get f of point five. So this would give you that lower sum, and that's an option one could choose. But on the other side, you could do the upper sum here, which the upper sum will always be choosing the maximum value that's in the interval. So for the maximum value, we're going to choose this one for the first interval, this one for the second interval, this one for the third interval, and this one for the fourth interval. So for your upper sum, you end up with one half. The delta x is unaffected by that. So you get f of negative one plus f of negative point five plus f of point five plus f of one. So it looks a little bit different when you get there, but the thing is the lower sum will always choose the minimum of the interval, and the upper sum always chooses the maximum of the interval. And you'll also have the effect that the upper sum is always, always, always going to overestimate the function's area. And the lower sum will always find the, it always gets smaller than the area under the curve always. And so the thing is if you know you're always going to get above and always going to get below, you can average those together, kind of like the trapezoid rule with the left and right rule. And you can use those to get a pretty good estimate when you average those things together. Now, this might not be the most popular technique that's under the sun, because in order to do these, these upper and lower sums, basically for each of these intervals one, two, three, four, you have to solve an optimization problem like we did in chapter four, which might not be the most popular thing to do. This example wasn't so bad because as it was decreasing for the first half and then increasing after that, it was pretty easy to determine where the maximum and minimums were. But for more curvy function, this can be much more of a challenge. But like our other examples, we can come over here to the web, and there is a nice calculator we can use here. This is a third calculator I want to introduce. The link you can find in the description below. The previous video had some other calculators you could use as well. This one is from MathWorld, which is sponsored by Wolfram. These are the people who make Wolfram Alpha. You can see the link to the right there. Also, Mathematica. And so let me make this a little bit bigger for everyone to see. Again, this will look much better on your own screen here. If we were to look at this function one plus X to X squared, we want to go from negative one to one. And we were using four rectangles, right? This here, if you choose the samples, it'll do left, right, midpoint. This calculator doesn't offer the Trapezoid Rule, which isn't too hard because you just averaged together the left and the right right there. But this one also does the upper and lower sums that are talked about in Stuart's textbook here. It calls them the maximum sums and the minimum sums. So if we choose minimum and we replot, it will change the picture over here, right? You can see there's the four rectangles. We did minimum, so we want the lower sum. So the height was determined at point negative one half, zero, zero and positive one half. And that gave us a value of where does it hide the number? Oh, it's in this yellow box right here. It gives you 2.25. You can ignore this actual area. That's the true area of the curve. We'll talk about that a little bit later. So it gives us the lower sum is 2.25. If we tried this again with the upper sum, and I do apologize for the size right here. The upper sum you gives you is 3.25, right? And so if you average those together, 2.25 plus 3.25, again, average that you're going to get about 2.75 as your average, which if you look at the true area of the curve, which is two and two thirds, 2.75 is actually pretty close. It's a pretty good estimate using just four rectangles right there. So again, I'm not going to ask a lot about the upper and lower sum, but if you do want a calculator, the Wolfram calculator from Mathworld does pretty good because you can do the upper and lower sum. The other calculators I showed last time didn't use those at all.