WEBVTT
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So for this question, we want to find the
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volume again. I'm pained by routine That region,
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bounded by X equals two times the square of why
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X equals zero. And why equals nine about rotated
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about the y axis. So once again, we
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want to start off by drawing region. So the
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X equals zero. It's just the Y axis line
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. So we don't really have to care about that
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, though. Why you called nine. We can
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just call it like that. Now the X equals
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two times the square off. Why so typically the
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square of why looks like this car shape are this
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conscience. Um, but the two times the square
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, why basically what it does is that it basically
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makes it wider, So it's still the same shape
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. We don't really care about what it specifically looks
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like because this is just a rough sketch. So
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if we draw a cross, the y equals nine
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line, we'Ll see that this is the region we
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have Ro Turner about the X axis is about the
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y axes. Excuse me. So once again you'LL
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see that we are actually making circles. So this
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brings us to the cross sectional area function. And
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in this case, it will be in terms of
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why equals pi r Squared R is equal to two
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because in this case are equal to the axe value
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. So you can just put that in. And
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so therefore a Why is equal to pi times For
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why, now that we know that we can go
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into the interval itself? You look back, you'LL
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see that we'LL be integrating from y equals zero to
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white nine and we just put in the previous cross
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sectional area function into the interval. It's mine.
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It's two five. We can pull out the four
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pine because that has nothing to do. There's no
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relationship with wine. It's it's so that's still from
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zero to nine. Why do you want? And
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so we know that the anti derivative off Why is
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just y squared over two? So that's four pine
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. Why squared over two from Sarah tonight? Now
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, by the fundamental theorem of calculus, we can
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just plug this in. In this case, the
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zero doesn't matter because zero squared over two is just
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zero, so we just need to evaluate it At
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y equals nine equals four pine times nine square is
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eighty one over two. Thes to cancel out this
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because of two. So two times eighty one is
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just one sixty two. So you get one sixty
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two pi. It's as your final answer for the
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volume.