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