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Problem 69 Hard Difficulty

A hole of radius $ r $ is bored through the middle of 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

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

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LD

Lencelord D.

August 25, 2021

Aren't the ends of the cylinder-like solid curved?

Video Transcript

in this problem. It is given that a whole of radius R is bored through the middle of this cylinder. Mhm Right. Of radius R. That this capital R is greater than our that is smaller at right angles to the axis of the cylinder. Yeah. Set up But do not evaluate an interior for the volume. Great. Mhm. So here we are given with this. We're asked to find We equals to integration R -R two. And what that where you will be instead of. So how we will calculate. Listen, let us see. So the largest gender access lies in the excess excess straight then Vice where plus, That's where it was to ask where that is capital elsewhere. And after that we will find that equals to plus minus or wrote over ar minus Why is square. Now this height will be one developing obviously yes because we are finding from the middle. Right. So that's where the height is. Double of the trade. No, the smallest slender exercise in that access then x squared plus y squared equals two R squared. Smaller square. That will be given by X equals to plus minus or route over small arms square minus y squared. So the land will be one length will be to root R squared minus one is right square And art is small. Right? This is for the smaller cylinder. Right? So that's where small artist taken. Right? So now the volume of the solid cut out is evaluated as value will be this integration Als hide improvement. We have calculated for largest lender is small incident. How can we write this letters? Right. This is us actually. So this is r squared minus y squared, right? And this will be to route over. I spread my this place but say so. This is actually which was asked and this was given. This limit was given. Anti diva was also given. So we only needed to find this value for this. Well, so this is it. I hope you understood the concept. Thanks. We want to.