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(a) Estimate the area under the graph of $ f(x) = 1 + x^2 $ from $ x = -1 $ to $ x = 2 $ using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.

(b) Repeat part (a) using left endpoints.

(c) Repeat part (a) using midpoints.

(d) From your sketches in parts (a)-(c), which appears to be the best estimate?

a. base 0.5

b. $0.5[ f(-1) + f (-0.5) \cdots + f(1.5)]$

c. $0.5[ f(-0.5) + f(0)+ \cdots + f(2)]$

d. $0.5[ f(-0.75)+ f (-0.25)+ \cdots + f(1.75)]$

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Missouri State University

Campbell University

Harvey Mudd College

in this problem, we are talking about the Rheiman some definition of an integral and how to determine which a method or really how any number of rectangles we pick gives us the most accurate description of the area under a curve. And that's what we're going to be talking about. And essentially comparing um two different number of rectangles in this problem. So the first thing that we're given is our function itself were given F of X equals one plus X square. So for part A this is just a general sketch of what we're seeing. This is a very crude sketch. Um So maybe you want to graph this on your own time if you want to, this does show us what we're essentially looking at. We have six rectangles under our curve and most of them fall above our function, some below. Now we're going to have to delta X values one for the three number of rectangles and one for our six rectangles. So for delta X we're going to have to plus 1/3 which is one. So the width of our rectangles is going to be one when we have three of them. And then similarly our Delta X. or the width of our rectangles will be .5 and we have six rectangles. So now we want to find the Estimations for R. three and R. six. So all we're doing is finding where the function or where the rectangle hits the function and multiplying it by delta X. So our three is going to be one plus two plus five which is eight. Then we do the same thing for our six and we get 6.875. Now for part B we're doing the exact same thing but now with the left end points so L three would be equal to two times, pardon me, two plus one plus two which equals five. And then we're going through the exact same process for L. Six, find where the rectangle hits the function and multiply it by delta X. And when we do we get 5.375 Now for part C, we want to estimate the midpoint. So now instead of taking the extreme A the left and right endpoints, we want the middle value. So M. three. The mid points for our three rectangles Would be equal to 1.25 plus 1.25 plus 3.25. And we get 5.75. And then similarly we're going to find the exact same thing for M6 and we get 5.9375. So these two sketches are very very crude sketches of what these mid points are representing. We have three rectangles, were taking the mid point, the middle value of that rectangle, and we're also doing it for six rectangles. So now the last part of this problem is asking, well, which estimate is the best one? Well, that's going to be M6. The midpoint was six rectangles. Now, why is that? Well, number one is we are considering the midpoint, so we're finding the average between the left and right endpoint. That's going to give us a better interpretation. And then we have more rectangles and what we should know is the more rectangles we have, the closer we are to the actual area. So that means M six. The midpoint of the six rectangles is our best estimate for our function. So I hope that this, pardon me, I hope this helps you understand a little bit more about the rhyme and some definition of an integral and how we know which method being left, right endpoints or the midpoint is going to be best for the estimation of the area under a curve. Okay,