WEBVTT
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find value given the equations. Why equals Ellen of
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X? Why equals one? Y equals two x
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equals zero, and we're rotating around the Why access
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. So the graph for the natural log of X
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looks like a log rhythmic function and just gonna kind
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of estimate that at this point, obviously, there's
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not a whole lot of exact points on the graph
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of Ellen of X. But I do know that
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it happens at 10 Ellen up one gives us an
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output of zero. So I do know that one
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good ordered pair here and then the rest of it
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, like I said, kind of just that log
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shape. So then let's fill in our vertical and
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horizontal lines as well. So the horizontal lines we
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have our why equals one y equals two. So
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why it was one y equals two. Well,
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you know that's not horizontal better. And then the
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vertical line X equals zero. So if you look
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carefully, the area that's bounded by those four things
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is actually this area here between one and two between
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X equals zero and then the function Ellen of X
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. Now we're going around the y axis. So
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as I look at that shaded area to the y
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axis, there's no missing pieces, right? All
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comes up front right next, it right next to
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it. Which means that is a disc method.
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So for the disk method, your value is pi
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r squared in the in a girl and you use
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whatever variable your revolving around this case D Why?
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Because I'm going on the y axis, So I
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need by y one and y two, we already
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have our Y values based on the boundaries. 12
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So then the only other thing we need to fill
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in is that radius that are squared is tell me
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the radius of the figure that's being revolved. The
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radius obviously goes from the center to the edge,
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and the center is happening wherever you're revolving. So
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here the Y axis is my center. The furthest
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out I go is to the graph of Ellen of
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X. So for the distance, for our I
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would think about what's my right minus left function,
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which would be Ellen of X minus zero, is
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the y axis. But look, if I feel
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that in what's wrong here, I don't have my
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variables the same. I can't integrate X with respect
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to why and expect to get the correct answer here
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. So I do have to change this to be
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in terms of why So I need to solve Why
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equals Ellen of X for X equals a function with
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respect to y. So to get X by itself
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, we need Teoh use e remember he rewrite the
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natural log. It's saying e toe what exponents gives
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me x e to the y gives me X So
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this is actually my function solved for X equals.
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And that's what I want to fill in here as
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my radius because you want to use the variable.
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Why so e to the y is what I wanna
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have here. This in a girl is a pretty
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easy you substitution. If you re right e to
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the y squared as each of the two y and
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then you can make to why you're you which means
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the derivative your do you is to de y so
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rewriting this integral would be 1/2 e to the you
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Do you so like I said, pretty easy to
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go through and solve it by hand here, obviously
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, if you have a calculator you can go through
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and just calculate it. Once you're at this step
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, but finishing out the problem, then your entire
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derivative of each of you is eat of the U
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and I'm gonna go ahead and change it back to
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wise. So I get 1/2 chi e to the
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fourth minus 1/2 pi e to the second filling in
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upper bound, minus lower bound. This is a
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totally acceptable answer, or you could factor out what
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they have in common. They have a 1/2 PYY
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squared in common, which doesnt really make it look
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any nicer. But it's a factored form of your
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answer. Both are equivalent. Both are acceptable for
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sure.