WEBVTT
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my were given a solid S and we're asked to
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find its volume. So we're told that s.
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Is a first strum of a right circular cone.
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There's a nice argument. Finally, there is a
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city party. I have just moved to loser indians
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with a height of each, a lower base radius
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of big are. And now I find that he's
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not here. Oh damn. My episode, freezer
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and a top radius of little art bananas can freeze
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. So help out straw a quick sketch. So
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we'll draw a cone along the X axis. Don't
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freak out. So we'll have uh the narrower part
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of the cone near the y axis in a larger
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part of the cone further away. Oh you were
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still in a crispy and shit, I still wanted
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crispy. I like a cold fruit. We know
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that this bigger radius is are, the smaller radius
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is little are. And the height distance between these
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two points is h have you ever had the gods
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go fuck you know, I just say yeah take
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positions, chameleon number. Yeah. Which is the
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volume can be found using the disk method like 30
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minutes. And I'm like well the problem with freedoms
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this disk has a radius which is well to find
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the radius. Yeah I get slippery. They're given
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by the line connecting the 0.0 R. And H
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. Big are so to find this line zero little
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R. Two H. Big are well we have
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a Y intercept of little are so Y equals Mx
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plus little R. And plugging in. We get
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big are equals M times H plus little are so
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M. Is equal to big Ar minus little are
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over H. And so the line has equation Y
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equals big Ar minus little are over H times X
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. Plus the lar and so the radius is big
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. Ar minus little are over H. X plus
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little are so I guess it's just steve hard.
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And now with our radius we use the disk method
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, their volume is given by pi times the integral
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from X equals 02 X equals H. Of our
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radius which we saw was big. Ar minus little
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are over each times X plus little are squared dx
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. Yes. Mhm. To solve this integral actually
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make some substitution is to simplify. So I'll let
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you equal big ar minus little are over H.
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X. Plus little are and we have the D
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. U. Is equal to big ar minus little
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are over H. Dx. And so we get
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that V. Is now equal to pi times the
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integral from we plug in zero. We get little
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are two plugging in. H. We get big
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are Where'd you get that? He's like I got
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in queens, my cousin. He's uh he's in
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eyes so I can't imagine. Mhm. And this
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now becomes you squared. Then dx is equal to
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H Over big ar minus little are do you industry
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in phase of his career? That's just this is
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a much simpler integral to evaluate. This is pi
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H over big Ar minus little. Our times one
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third U cubed from U equals little are too big
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are. Yeah. Tonight. And this is equal
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to pi H times big R cubed minus little R
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cubed Over three times bigger. R- Little are
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. Now we can factor the top using difference of
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cubes. This is pi H times bigger R minus
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little are, time is big. R squared plus
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no gays. Big r little R plus little R
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squared Over three times bigger R- Little are.
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And cancelling out they're upstairs. Their blood is we
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get pi H over three times big R squared plus
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big are times little R. Plus little are squared
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. Um On a day where you're sure these days
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, weeks like