WEBVTT
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we want to find the area of the region enclosed
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by these two functions. The 1st 1 in blue
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Why equals three X squared minus two X and the
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second in green like was X to the three minus
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three X plus four. So if we do a
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quick graph using a graphing calculator, at first glance
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it looks like there's 11 region over here because,
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uh, at first glance it looks like there's two
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intersection points. But if we actually zoom out,
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we find that there's 1/3 intersection point which happens Ah
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, at a higher X value. Um, so
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we actually have two regions, the 1st 1 being
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down here, which we saw in the first slide
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. And the second region is this tiny sliver that
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goes over here. So the total area is actually
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going to be the sum of two areas which will
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solve independently. So all right, that this as
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area is equal to area one plus area two.
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So we're going to figure out area one in area
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to separately. So area when I do it in
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green and area in red. An area, too
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. I drew in black, so let's all these
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two areas separately. Area one is equal to We're
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gonna do an integral So let's look at our first
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picture. Since it's a bit more clear, our
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first X value is going to be negative. 1.115
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on our upper value is going to be 1.254 So
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let's write those in negative 1.115 to 1.254 and it's
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going to be the upper function, minus the lower
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function. Here. The upper function is the green
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one, and the lower function is the blue one
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. So we're going to subtract those the green function
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executed minus three X plus four minus the blue function
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three X squared minus two X dx. Okay,
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of course, we can simplify this a little bit
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. We haven't x cubed. We have minus three
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x squared we have in minus X and then we
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have a plus four DX so we can carry out
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this into girl by doing the anti derivative 1/4 ex
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to the fore minus X cubed minus half X squared
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plus four x. And then here we're plugging in
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1.115 and then 1.254 So plugging this in gives us
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approximately 6.185 as area one. So now let's determine
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area too. Again, we're doing an integral So
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now we want to look at this is Uno graph
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. We see that our lower ex limit is 1.254
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on our upper X element is to point X 61
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So let's write that wrote 1.254 Uh and our upper
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limit was 2.861 So now we're doing our upper function
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Minus are lower function. In our first integral,
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the green function was on top. So in this
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region, the green function is actually on the bottom
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. So it's going to be our blue function minus
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our green function. So let's right there. We
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have three X squared minus two x minus X cubed
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minus three X plus four D x. Okay,
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so this is actually just a negative of our first
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integrated are the Inter Grant from Area one. So
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let's just use that we're going from 1.254 Sorry,
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there's no minus here. 22.861 and the integral and
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here is just going to be the opposite from area
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one. So we have in fine negative 1/4.
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Oh, sorry. Glad to Reese this. Let's
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just simplify it. We have a negative X cubed
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minus three X squared, minus X plus four D
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x. And then we just, uh, carry
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out the intern. The integral negative. 1/4 next
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to the fore, plus x cubed plus half X
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squared, minus four x hopeless works. Oh,
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sorry. This minus. I just popped in minus
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four X. There we go. And we're going
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from 1.254 22 points. 861 Plaguing. Our numbers
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in gives us approximately 2.193 And now to get the
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total area, it's area one plus area two,
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which is approximately 6.185 as area one and 2.193 as
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area, too. Which gives us our approximation for
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the total enclosed area. 8.378