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Find the exact area of the region under the graph of $ y = e^{-x} $ from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b).

$\int_{0}^{2} e^{-x} d x=\lim _{n \rightarrow \infty} \frac{2}{n} \sum_{i=1}^{n} e^{-(2 i / n)}$

01:32

Frank L.

Calculus 1 / AB

Chapter 5

Integrals

Section 1

Areas and Distances

Integration

Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

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Yeah 5.1.30 You have a function y is the exponential. E to the minus X. Were interested in the domain of this function From 0 to 2. So if I take a look at what this function is doing zero, it has a value of one and it too has a value of one over E squared. So somewhere around one of one third. So it's curve is doing something like this. So if I use rectangles to approximate the area, you're going to see this is going to give me the area under the curve above the X axis. So let's use in rectangles. Yeah that means each rectangle has a width Yeah of 2 0 over in which is to over in I will obtain the height of each rectangle by evaluating the function. So this first one is going to be E to the um that's going to be if the first one is going to be incriminating it by two over end. So that's one times to over in. The next one is going to be at E to the excuse me as a minus minus two times two over end and then the last rectangle you're going to get is going to be E to the minus and that's going to be um in Times two over in or eight of the -2. So if you write this out as the number of rectangles goes to infinity, that's the limit as N approaches infinity of the width of each rectangle. Yeah. Now some up each of the heights, so from my equal one to end it's E to the minus and that's going to be To over n times I or -2 I over in. So this is going to give me the exact area under that curve. Now let's go to a computer algebra system to figure out these different pieces. So if I come in here I find out that the sum so two of her in some I go into in an E. To the minus two I ever and it's two E squared minus one. So this is going to be the limit as N approaches infinity of two. He squared -1 over and that's uh any squared, so in he squared each of the two over in -1, and I think I've got that correct. Yes, so that gives us the correct value there, then what we want to do is from there. Yeah, yeah, evaluate this limit. So to evaluate this limit, wow. Mhm. Mhm. Go back to our computer algebra system and let's just evaluate what this limit is as n goes to infinity And you get your value there one minus one over E squared. So this turns into 1 -1 over he squares. So that tells me the area is one minus one over E squared

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