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Numerade Educator

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Problem 3 Medium Difficulty

(a) Estimate the area under the graph of $ f(x) = 1/x $ from $ x = 1 $ to $ x =2 $ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Answer

a. $\frac{533}{840}$ underestimate
b. $\frac{319}{420}$

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Video Transcript

and this problem, we are understanding the rhyme and some definition of an integral. Where specifically we're looking at a function and finding the area under the curve by looking at an infinite number of rectangles that make up that curve. And what we're essentially doing is taking a some of the area of every single rectangle until we get really, really close to the actual area. So that's what we're talking about in this problem. Were given the function F of X equals one over X. And we need to estimate the area using the right endpoints and the left end points. So for part A we have to look at the right end points of our function. The first thing that we need to know is delta X. What is the width of these rectangles? Were even looking at? Well, it's going to be be over, pardon me, b minus a Oliver end. So that's two minus 1/4, which is 1/4. So this is a quick sketch of our function. This isn't perfect, but this dust does give us a quick representation of what we're doing. We have four rectangles underneath the curve and you can tell these rectangles are not going above that line. So this is probably going to be an underestimate. So now we have to use the definition of the Riemann sum. So are four is going to be equal to the sum from I going to 1 to 4. So that's just the number of rectangles of F. X. Of I times delta X. Essentially what that means is we're taking the point of the function where the rectangle hits times the width of the rectangle. So this simplifies to 1/4 times this entire portion 4/5 plus 2/3 plus 4/7 plus one half. And that simplifies 2.6345 And this is going to be an underestimate for the area. Now, for part B we're doing something very similar, but now we're taking the left end points. So again, this is a quick sketch and now you can tell our four rectangles are above the curve, so this is probably going to be an overestimate, so we're going to do the same thing, we're going to find the rhyme and some, so L four is going to be equal to 1/4 times one plus 1/1 10.25 plus 1/1 0.5 Plus one over 1.75. That simplifies 2.7595 and that's going to be an overestimate. So now what this essentially telling us is we have the right endpoints and the left endpoints wants an overestimate. Want an underestimate. So the true area of the function is going to be somewhere in the middle, and that's where we would use the actual algebraic integral to tell us the exact area. So I hope that this made sense, and I hope you now understand a little bit more about the Raimund definition of an integral.