WEBVTT
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Let's find the volume of the solid obtained after rotating
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the region underneath his curve from zero one. So
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here's a rough sketch of the graph when you plug
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in X equals zero to get one over two.
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As Ex increases, the denominator is larger, so
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the fraction is a whole good smaller. That's why
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it's decreasing. So for part A, we wrote
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it about the X axis, so basically we could
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see that a cross section is a disc. So
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we used the volume, so we know the disk
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will have radius pi r squared. That's a circle
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right there. So the radius is just the distance
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from Y equals zero Up until the curve that was
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in a straight line. Let me draw back here
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. So there's the radius from center to the end
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point. So our is just equal toe Why,
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which is given by this? So that's the radius
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of the disc. So for part, a pull
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off the pie that's coming from a circle area and
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then here's R R squared. So that's the formula
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for the volume using the disk method. So we'LL
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just have to evaluate this. Let's go ahead and
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factor that denominator before we do partial freshen And now
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we'LL just spend a few moments simplifying the inter grand
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so we can write this over explicit one. Be
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a restless once where c over X plus two and
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then be over X plus two squares. But so
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I'm running out of room here. I'Ll need to
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go to the next page, But before I do
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, let me note that for part B, when
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we rotate this thing about the Weisses, it may
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be easier to use cylindrical shells then to use this
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method. So for part b of hunger or use
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the shaman okay, simplifying our pre use equation.
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So we multiply both sides by the denominator on the
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left, clear out the denominator and then multiply out
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the right hand side and we'll have a system of
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equations coming up. Surely so here we can expand
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the whole thing on if we want to find a
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, B, C and D, or you could
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just go ahead and plug into Mike's values. For
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example, if you plug in X equals one,
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the left hand side is just one. We're on
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the right hand side. We should get in this
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case. I shouldn't be plugging in one. Let
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me plug in negative one. So if we plug
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in that we have zero over here, a zero
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over here and also zero over here. So then
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we just have be on the right. So that's
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one value. If we try, X equals negative
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too. We'Ll get d equals once. Now we
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could use some reasoning to find the other variables here
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. So, for example, if we look at
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the coefficient of that's cute on the right hand side
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, we see that there's two exes here, another
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X here. So that's one for the a term
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and that also see, we'LL have an excuse to
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access here Another meds here. So a plus e
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must be zero that is, this equals negative c
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and then looking at the constant term on the left
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, we have won on the right we have for
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a plus four b plus two C plus the yeah
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, plug in our values for being D and then
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also replaced, See with minus a and then So
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go ahead and solve this and we get that a
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equals two. Excuse me is minus Who so that
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c equals two. Now it's God and plug in
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these values for a, B, C and D
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and Tara. Partial fraction to composition. So we're
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still in party here, but we're basically finished.
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We've simplified our into grand, and now it's going
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to be simpler. Okay, there's a partial fraction
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to composition that they're plugging in are constants Now.
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We could go ahead and integrate all of these.
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The first general, if that plus one, is
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bothering you. Feel free to deal use up here
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. Same thing for the Senate next in a role
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as well, but here and could use the power
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rule. It's plus one. Similarly, here you
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have natural log and then power once more and then
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zero to one. So then which is good and
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plugging those end points? And I should have wrote
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. After that, there was a pie appear.
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A part of me should be a pile here the
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whole time. Now let's just go ahead and plug
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in those in points. So that's the plugging in
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the one. And then now we plug in zero
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Ellen of one zero. So that's what we get
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after we plug in the second and point and then
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just got and simplify this. Use your properties of
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the lot of them and we get in. Answer
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that looks like this. That's for answer for party
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now, as I mentioned before, you shells for
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part B after rotating it about the y axis.
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When I do so, my formula using shells too
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high in a girl zero one Then I have X
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and then on the denominator. We've already factored this
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. So let me write. This is X Plus
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one explodes too. This is the original denominated from
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the original function. Now let's do a partial fraction
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to composition. Here we can rewrite this using what
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the author calls case one distinct linear factors. Then
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you could multiply both sides of this equation here.
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By the denominator on the left, X equals a
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X plus two plus B. It's plus one.
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So here the partial fractions does seem a little simpler
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than party. Last time we had four coefficients.
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This time will we have to. So we see
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this has to be one that has to be zero
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. So go ahead and solve this for A and
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B. When will plug in these values of Andy
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into our partial fraction and then we'LL go ahead.
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And Mina Marie. So this all becomes plugging in
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negative one slugging. You're being equals two. Go
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ahead and evaluate these intervals where those will give us
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natural logs using you. Slip if you need to
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. And then our one point zero one. Now
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go ahead and simplify this. And then let's go
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ahead and finally break This's to pie. Natural log
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nine over eight And that will be our final answer
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for part B using show mother.