WEBVTT
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for this problem, you need to be looking at
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the book. It's referencing a certain region and a
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certain line that we're going to be looking at as
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we generate a volume. So for this problem,
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we are looking at Region One. So that is
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the area. I'm going to try to keep the
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same color scheme as in the book. That's the
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area under this line. Okay. And what am
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I going to do with that? I'm going to
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rotate it around the line B C. Well,
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BC is Y equals one. Okay, because this
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is B. This is C So we're rotating it
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around. Why equals one? So let me just
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put this on here. There's my rotation. So
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as I rotate this and I generate a solid let's
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see what we have. If I just take a
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general piece, just a random piece right here,
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Representative slice, This is going to be a washer
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because I'm going to have an empty space where that
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white is. And then once I hit the blue
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, I'll have that as it goes around. This
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is a washer. What is the volume of a
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washer? Well, a washer is going to be
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two circles. First I have the outer circles.
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I'll call that my big our pi r squared minus
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the inside circle, which is pi little r squared
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times the height. If I want, I can
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also pull the pie out. I could say this
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is pi times big R squared, minus small r
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squared times the height. That's how I find the
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washer method. Or find the volume using a washer
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method that's each individual washer. I could use that
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formula to find the volume, and then I stack
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all the washers up. So let's see what that
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looks like when we use calculus to solve an infinite
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sum of an infinite number of washers, all stacked
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up. As I stack these washers, I'm going
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to be going from X equals zero two x equals
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one, which means I can pull the pie through
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. I'm going to have the ours, which I'm
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going to look at in just a minute times the
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height and the height is that little infanticidal change in
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X or D x? That's that becomes my height
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. So what are my two radius is? Well
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, the largest radius goes all the way down to
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the X axis. It's one, and it's uniformly
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one all the way across this distance. So my
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capital are the big radius squared is just one.
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So what is our small radius? Well, the
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small radius and I'm going to put this in green
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here. Is that line going from the origin up
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to Point C, which is 11? Well,
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if I think about what the equation of that line
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is, this intercept is zero. The slope goes
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from 00 to 11 The slope is one or the
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equation is just y equals X. So as I
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go on that, um, my representative disk here
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and I go down, I hit that green line
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. That's the line Y equals X. So that
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radius is X and I'm squaring it. So this
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is based on our washer definition. This is the
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setup for the integral that we're going to take to
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find the volume of that blue section rotated around the
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line Y equals one. So let's actually do the
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calculus are the integral of one is just x the
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integral of X squared. What we add one.
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And we divide by that exponents and we evaluate from
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X equals 02 X equals one starting at the top
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limit of integration. I get one minus one third
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and I would subtract the lower limit of integration.
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But that's just zero. So I don't have to
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worry about that in this problem. So this is
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just two thirds times pi or to pi over three
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.