WEBVTT
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it seems like you guys fell asleep so I might
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want were given a solid S and rest to find
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the volume of this solid, we're told that the
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base of S. Is a circular disk with a
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radius. Little are that parallel cross sections perpendicular to
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the base? Are squares look well. But is
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it so the equation of the semicircle above the X
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axis? This is why equals the positive square root
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of little. R squared minus x squared. And
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the equation of the semicircle below the X axis is
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Y equals negative square root of R squared minus x
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squared. These bounds the base and therefore we have
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at the side length of the bottom of the square
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, which I'll call S. This is our upper
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value of why? Which is the square root of
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R squared minus X squared minus the lower value of
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why? Which is the opposite of the square root
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of r squared minus x squared which is two times
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the square root of r squared minus x squared.
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Right? Yeah, all turns out so this is
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the length uh side length of the square pointing out
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of the circle. Yeah. Now define the volume
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. We should find the area of one of these
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squares as a function of X. Well, I
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was well this is the area of a square with
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a side length and so it's going to be the
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side length two times the square root of r squared
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minus x squared squared, which is four times r
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squared minus x squared Israel to. And now we
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have the volume using the cross section interpretation. This
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is the integral from X equals negative art positive our
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of our area A X of the squares pointing out
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of the circle, D X. This is the
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same as four times the integral from negative are positive
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. Are of r squared minus x squared. The
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X. I can imagine this is a pretty simple
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integral to evaluate but I'll walk through since we do
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have some and all the variables in this. So
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this is equal to take the pussy on four times
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R squared x minus well mr Niva. And this
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is X. To the just letting us know that
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Science third power over three from X equals negative are
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two positive are and if you plug in well we
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get four times To r cubed over three. It's
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named that wasn't there bitch? It rode a horse
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naked lady which is equal to she really she's not
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very good doing china. He rides around yes minus
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four times negative. Two. R cubed over three
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. You can see it. This is equal to
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is sloppy eight plus eight or 16. R cubed
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over three. Good job. Really good.