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Problem 23 Medium Difficulty

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_2 $ about $ OA $

Answer

$\frac{1}{3} \pi$

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Video Transcript

okay. We want to send this green region about the line. Oh, a which is the X axis there. And we want to find out. What's the volume of this shape that we get here? Okay, so we're going to cut perpendicular to the axis of rotation. So this way, So you can see that when I get there is a disk with a hole in it. Okay, So washer can the way to find the volume of that is pie bigger r squared my little r squared. H, We need to find out what are R and h r. Okay, So h is the height of the cylinder or the disk, which is this which on here is from here to here, so we can't see that's DX. Get a little ours from the axis of rotation out, so you can see that that is some Why value. And in this case, it's why equals the fourth root of X little are is why the 4th, 3rd of X and then big ours from the axis of rotation out to the outside edge. And you can see that this one doesn't change. It's equal the one all the time no matter which one I draw. So that's why equals one. Okay, then we're gonna pile these little guys up from here. X equals zero thio here. X equals one. The volume is pie integral 01 for big are one squared minus little are fourth root of X squared times H d x tau pi 01 one minus ok. The fourth root of X is the same thing as X to the 1/4. So the fourth root of X squared is X to the 1/4 squared. That's X to the tooth force or X to the one half. All right now in a great hi x minus Extra three halves over three halves 101 X minus two thirds X to the three halfs was your one coach So pie times one minus two thirds minus zero minus zero. So one third pie