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If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integrals in Exercises. Use $n=4,10,20,$ and 50 subintervals of equal length in each case.$$\int_{0}^{1}(1-x) d x=\frac{1}{2}$$

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Definite Integral

Integration

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Okay. What we want to do is we want to use a calculator or a program that will draw some agreement re Monson's for us, and we want to show that the integral from zero toe one of one minus x t x converges to 1/2 and we want to do it for in equal four, 10 20 and 50 sub intervals. And so what we're gonna do is I'm going to switch to, um, a graphing utility, um that is capable of drawing those rerun sums. And so it's Dez mo's. Um, if you have a t I 84 there is a special program that you can actually download. Um, that will also do this for you. And so we want to First of all, um, changed this to, um one minus X. Um, and so it's a line man are left in point on the integral waas um zero and our upper limit waas one. And the number of intervals is for the first go round is a four. And so that's what the drawing would look like if I had four rectangles. Um, and this is doing the left in point some. And so, um Yeah, and you could change it. This is a pretty interesting one, because you can change it to Dio. And this would be for a decreasing function. The left in point would be the upper some, and if I change it to the right in point, it would be the lower some in this case, we're just gonna keep it said at the upper some, um and then also this allows me to also do the midpoint. And so we're just gonna keep it at the left, handsome and for the left hand sound for decreasing function is going to be your upper limit or your upper sound. And then the integral that we get when we do the value of the integral when we do for four rectangles is 40.6 to 5. Um, which is not close to 1/2 but it's getting there. And so what happens now if we just change the number of rectangles to 10 7 novels? And so now we're doing 10. And so you notice now that the value the integral role goes to 0.55 Now, what happens if we dio 20? And so here we have 20 you know, girls and we get 200.5 to 5. So it is converging or it is approaching. The more as the partitions of the right tangles get smaller or the number of rectangles gets bigger, then that in a goal, is approaching 1.5. And so now let's see what it does when we do 50 rectangles. Yes, point Dr One. And so it is approaching 1/2. So that's what we're doing for this problem. Dez knows. It's, um, free. You just have to sign up for it. Or like I said, if you have a graphing calculator and inspire or 84 I know the 84 you can get a downloadable program to do the exact same thing.

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