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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.$y = \sqrt{x - 1}$ , $y = 0$ , $x = 5$ ; about the x-axis

## 8$\pi$

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Applications of Integration

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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### Video Transcript

chief. We are given curves bounding a region and align and we're asked to find the volume of the solid obtained by rotating this region about this line. Then rest to sketch the region. The solid and a typical disk or washer. The curbs are y equals the positive square root of X -1 And y equals zero As well as x equals five. And the lime rotating is about the x axis. So first I'll actually sketch the region along with a typical cross section. So it's bounded by a square root function as well as horizontal and a vertical line. It looks something like this. Yeah. Yeah. So our square root function related and begin until x equals one. We go away from 1 to 5. Yeah. He's founded by square root function but also the horizontal line, Y equals zero. In the vertical line X equals five. It looks something like this. Yeah. So this in green is our region are to sketch the solid nine inch. You want to rotate about the X axis. So it looks a bit like this. Four brought your ass off. I was portion. Yeah. Does that mean he wants a car? So this area and this solid and blue. This is two solid and a typical disk. Well looks like this sneakers. He said great tick. That's how much time you wasted. Not doing homework. That's the sound of the clock. Chicken on your ass cops. Just it's not a watcher. There's no hole. Now it follows from the disk method that are volume is going to be the integral from X equals 12 X equals five of pi times our radius which is the radius of this disk which is square root of x minus one squared dx. So with a kiss, this is equal to High times the integral from 1 to 5 Of X -1 DX. This is pi times one half X squared minus X from 1-5, plug these in. This is equal to eventually you get eight pi. Just another guy comes in. Score myself, I think this one.

Ohio State University

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Applications of Integration

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