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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = 2 \sqrt{y} $ , $ x = 0 $ , $ y = 9 $ ; about the y-axis

162$\pi$

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 2

Volumes

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

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