WEBVTT
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were given a solid S. And we're asked to
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find the volume of this solid. We're told the
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base of S. Is a triangular region with overdoses
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. Who cares? I don't know. There used
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to be 00 10 01 In the cross sections perpendicular
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to the Y axis are equal lateral triangles. It
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might help to actually draw this. So we have
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our X. And Y. Axis. Uh huh
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. The river is is at 00 10101. Yeah
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, suck my dad. And then we have cross
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sections perpendicular the Y axis that are equal lateral triangles
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. Look something like this. Imagine is coming out
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of the plane of the page. Now mm He's
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just he's rich now you know he's still following him
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. Hell yeah. Which? No I guess headache
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. We want to find the area of each of
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these equilateral triangles. To do this we need to
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find the length of a side of an equilateral triangle
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. And to do this, redefined what curves bound
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this red region. But we have the curves X
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equals zero of course and Y equals zero. And
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then we have the line between the points 01 and
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10 This is the line Y equals negative X.
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Plus one. And therefore the length of a side
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of a triangle which I'll call s is simply uh
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we can solve this curve for X and get X
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equals one minus Y. Not that you can keep
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came out. Yeah. Yeah. And it I
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don't I don't remember. We are just like yes
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, who's the new Orleans guy with that song about
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? I'll be fine. It sounded in my mind
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that because it was a thing where it was like
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there's another song called Let Me Find Out By is
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that Yeah, the Fifth World. Therefore we should
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be Let me find out your old man. We're
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saying that was that's a sign that something Mark,
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find out your families who have it's been me.
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Right? So the side is actually Uh just one
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-Y. And therefore the area my notes where there
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was between the notes of the equilateral triangle. This
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is going to be one half times the base which
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is one of these sides, times the height.
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And for an equilateral triangle skyscraper, we divided up
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into a right triangle. Like this, we have
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S over two squared plus the height. H squared
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equals S. Squared. So that H. Squared
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equals a nancy. The connotation 3/4 S. Squared
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and therefore H. Equals Route 3/2 s. And
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so we have one half S. Times route 3/2
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S. And so as a function of why the
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area A. Of Y is one half times one
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minus Y. Times route 3/2, of course times
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one minus Y again, which is root 3/4,
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1 minus y squared and therefore the volume is the
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integral. Overall the triangles. This is from y
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equals zero To y equals one of the area A
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. Of Y. Yeah. Dy well, two
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. So this is route 3/4 times the integral from
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0 to 1 of one minus Y squared. Dy
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. If you evaluate this integral, Eventually get root
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3/12 and voices.