WEBVTT
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We want to find the area enclosed by these four
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equations. So first, let's do a quick sketch
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to see what they with these equations look like.
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The 1st 1 is Why's the hyperbolic sign of X
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? Or some people say Cinch X? Um,
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the second equation is e to the negative X.
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Are these air just sketches? So they're not 100%
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accurate. Let's get the 3rd 1 For now,
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let's actually show the 4th 1 X equals two,
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which is somewhere there. And if we were to
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pluck from the 3rd 1 X equals zero Hey,
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we wouldn't have an enclosed region. So the way
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that this question is posed, there's actually a mistake
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. This should say why equal zero, which would
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give us this region from this line over here.
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And so we get this enclosed region here. So
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that's the area that we want to determine. I'm
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so of course we're gonna be integrating. So we're
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going to integrate from X equals zero two X equals
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two of whatever our top of function is on the
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region minus our bottom function. However, we do
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have this point over here where the top function changes
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so we want to figure out first. What is
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the X value here? X equals question Mark.
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So the way that we solve this is to equate
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the two equations that cross at this point the 1st
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1 beings sin checks and the 2nd 1 being e
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to the minus X. So let's just equate those
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himself for X meat to the miners X. So
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we recall that cinch X is e to the X
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minus e to the negative X and that over too
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on we create that eat a negative X and then
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we just solve for X so we can multiply both
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sides by to e to the X And in here
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we have e to the minus X and each of
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the ex canceling to give us one. So we
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have a two on the right side. We have
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two's canceling here and we distribute e to the X
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. So we get e to the two x minus
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one. So we bring this one over, we
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have a three and then we take the natural log
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a rhythm that's an X equals natural logarithms of three
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. So X is equal to ah half of London
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of three or bringing this half inside. We can
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write this as long of screwed three. So that's
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the point of intersection over here. So X is
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lunch of route three. So to integrate over,
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to integrate, um, to find the area of
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this region, we're going to writers to inter girls
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first going from X equals zero X equals one over
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three, and the second Integral is going to go
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from X equals a lot of room 32 X equals
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two. So let's write that up. Uh,
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area is equal to integral. We said X equals
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zero to London of route three. Um, and
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what we have to do is take the in the
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top function, minus the bottom function. Over here
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, the top function is cinch X, and the
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bottom function is X equals or y equals zero s
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. So we just subtract those cinch x minus zero
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. I I won't write the minus zero d X
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plus integral. We're going from lawn of Route three
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two X equals two. And so we have our
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top function minus their bottom function. Here, the
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top function is eating negative X, and the bottom
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function once again is y equals zero So we integrates
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e to the negative X minus zero. This is
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D X and then so we recall that the anti
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derivative of cinches, cautious or hyperbolic, co sign
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Kash X. We're going from zero to lawn of
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route three plus Ah, the integral or the anti
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derivative of Eton. Negative X is negative E to
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the minus X And here we're going from loan over
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three up to two. And so the second piece
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should be easy enough. But the first piece,
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we have to recall that caution X equals E to
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the X plus e to the minus X over two
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by definition, and then so plugging everything in here
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we have a route three plus one over route three
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all over, too. We subtract the one No
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, we add, um, we plug in X
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equals two so negative e to the minus two or
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one of re squared, Uh, and then we
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subtract. But then we have a minus on them
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. Another minus. So that becomes plus so e
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to the negative. Long over three will work out
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to be won over three. And after simplifying everything
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, we get screwed three minus one minus one over
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East Squared, which works out to be approximately 0.59