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With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $ n $, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $ n $ = 10, 30, 50, and 100. Then guess the value of the exact area.

The region under $ y = x^4 $ from 0 to 1

$\sum_{i=1}^{n} f\left(a+\left(i-\frac{1}{2}\right)\left(\frac{b-a}{n}\right)\right)$

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mm. So this problem, they gave us a function Y is equal to X to the 4th and roughly you should know that looks like a very steep version of a parabola From 0- one. And were asked to estimate the area. Um using rectangles, they ask us to use an equal 10, 30 50 and 100. And fortunately we are doing this with calculating tools instead of doing this manually. So if N is equal to 10, then on the interval from 0 to 1, the width of each rectangle is 1/10 and then likewise, if n is equal to 30 50 and 100, that's where we're going to come up with the width of these triangles, each getting smaller and smaller as we have more and more triangles. But rectangles to do our estimates for the area. So now let's do this if I'm using right in points, so if I'm using a right in point, So think about this on the interval from 0-1, The right in point. That first endpoint is going to happen based on this width. So in the first case that first endpoint is going to be .1 in the second case, when n is 30th 30? That's gonna be 1/30 and then 1/50 and so forth. Okay? So and then we just need to figure out that tells me the width and then evaluate the function to get the height. So we're going to do all of this on our calculator. So we're going to do first is let's store this function. So X to the fourth, I'm going to store that function and just call that F of X. Yeah, for that as function. Sorry about that. Okay, X to the 4th and then move the cursor down and then store that as F of X. And I'm done. So now I can evaluate F of any any value and come up with that. So I've stored f now I need to create a sequence to get those right in points. Okay? So I'm gonna index with I. And where does that first endpoint start when N is equal to 10? That first endpoint is at 1/10 So the first endpoint Is going to be at .1, so that's the first right in point. And then I'm gonna go, how far how far am I going? I'm going all the way over to one and I'm gonna increment that by .1. So that's going to generate all the right in points for the rectangles. So I need to store this, so I'm going to store this in the variable, we'll call it this all of our X. Values. So that gives me all of the X values I need uh in the case and it gives me 10 of those to make that work now. So that gives me the right in point of the right side of the rectangles. I know the width is 1/10. So to get all of these are gonna have to sum up. So I'm gonna have to have The width of the wrecked the triangle, the width of the rectangles, .1, so .1 times the sum. Okay, I need to add up. Well I need to evaluate F at each of those points, so F of the X values. And that will give me .25333. So that is my estimate when N is equal to 10.25333. So let's just write that down. Yeah. Yeah .25333. Now we're going to go back into the same thing when in is equal to 30 and so what does that change? Well when N is equal to 30 I'm going to change my sequence. Okay so let's just go back up here and let's just do that all over. So instead of starting at .1 is gonna be 1/30. Okay. Mhm. But that's gonna be where the first point and then it's also going to go to one but I'm going to increment by 1/30 and that should give me a longer list of values. Okay so now what I need is I need to estimate the area. The width of each rectangle now is 1/30. So what is 1/30? So one 30th times F at all of these X. Values Excuse me. The sum of all of those times, the sum of all of the f at each of the X. Values. Yeah .2170. So .2170. Is what I come up with their, Now let's do the same thing with 50. So I go back to my calculator and I said okay well what would the sequence looked like if I had 50? Well the width would start the first, the first rectangle would start at 1/50 I would go to one and I would increment in increments of 1/50 And that is going to be my new list of exercise. I should have 50 of those. And then now the width is 1/50. I'm is the sum of F. Of all of the X. Values. .2101. So point to one no one. Yeah. Yeah. Yeah. Now let's go to 100. So go back to my calculator, recall that sequence in command. That generates all of my in points. Now instead of using 50, I'm going to increment by 1/100. So that's my first rectangle And I increments by 1 100. Those are my list of 100 x values. And now I just need to figure out well now the width of each of these is one over 100. Yeah. And times the sum of F at all of these X values. .2050. .205.0 yeah. Yeah. So what we can see here is each of these areas, as I increase the number of rectangles, the number is going down, they say, guess what you think this is? Okay. So if you already guess, you know, is it going down? Um It makes a big step when I go from 10 to 30. And then as I add, rectangles is making smaller and smaller steps. So you could guess that the exact value Is maybe .20. Okay, by the way, it's moving there so you don't know that you'll find out in your subsequent section. That that's how you figure that out. But I would make a guess. It looks like maybe this is converging to 1/5 or point to.

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