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Problem 53 Easy Difficulty

A hole of radius $r$ is bored through a cylinder of radius
$R>r$ at right angles to the axis of the cylinder. Set up, but
do not evaluate, an integral for the volume cut out.

Answer

$$
V=8 \int_{0}^{r} \sqrt{r^{2}-z^{2}} \sqrt{R^{2}-z^{2}} d z
$$

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Video Transcript

{'transcript': "we have a cylinder of radius R big R. And I drew that opening, uh, on the X on the x axis is its axis. And, um then we're gonna drill hole through it of Radius Little. R and Z is its axis on. We want to find the volume of material cut out. Okay, so what I decided to do is to cut it this way into slices so that each slice is a rectangle. Looks like that. Okay. And you can see that the length of each rectangle is from Z on the frontier to Z in the back. So to Z no. And the width of the rectangle from here to here, two x. Yeah. Okay. So that the area of the slice equals two x times two z or for Z. Okay, Now, I want to make it into a volume, so I'm gonna give it a little bit of thickness, okay? And its thickness is some Why amount? Because why is my up and down access here? Okay, so now the volume, so area of top of slice. So the volume of the slides is the area for eggs e times the thickness, which is d y all right. And so then the volume of the whole thing is Adul the slices up for X D d y from Why? On the bottom From why On the bottom. Come on, picture. So why on the bottom here would be big are why in the bottom here would be a little r two. Why on the top which would be a negative, are toe toe are Okay, well, so you should notice there's a problem with that. And the problem is, I'm integrating with respect to why so I can have an X and Z in there. So now I gotta figure out how X and Z N Y are related. Okay, Well, it's just equations of the circles. So if I squished up the red circle or the red cylinder and squished it up, it would end up here in the y Z plane and its radius is big. Are sorts equations y squared plus Z squared equals big r squared thick is me Z is the square root of big r squared minus y squared. And then if I do the same thing with the little blue circle, I squish it up It ends up here in the X Y plane and its equation is X squared. Plus y squared equals little r squared. So X is the square root of little R squared minus y squared. All right, so now I can just put those in and let's put the four out in the front. Let's instead of going from minus Are You Are Let's Go from zero are so twice. Zero R x, which is R squared minus y squared little our times E, which was a big R squared minus y squared de y. There you go, and that will get you the volume of the stuff that's knocked out when we drilled the hole in there. I think it's interesting it doesn't have a pie in it. Just something to think about, okay?"}

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