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

$ \Re_2 $ about $ OA $

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

Applications of Integration

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Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

in this problem. It is asked that refer to the figure and find the volume generated by rotating the given region. About the specified line are too Mhm. About oh eight. Right? So Hill Washer method is used about by equals to zero. So let us let us take very simple method, right? Or to understand it. So how do we understand it? We will we want the volume generated by rotating the given. Right? So we are creating in this way. Right? So when it is related in this way that is about Y axis, right? So we can say that this is the area. Right? So all that area covered by we want for our two only. Right? So for getting her to we must take the whole area like this. Right? And the limits will be 10 to 14 X. Right from here. Because we are rotating in this way, this is about by access. So this will be work. The limit will be from 0-1. This we know. But for finding out the specified line are too right about the way we want our two regions. Right? So, if we want area for the art, so whole area that is we will be getting this is are we can see, we can assume it are right? So we know that if it is rotating, then the area becomes pi r squared. Right? So area covered with the pioneer square. And we will just integrated. So this week we want we will get the volume in this way. So the r value is fall for here. And after that you can see pirates square. Right? So this is subtracted, totally subtracted with this girl. Right? So the equation for this girl is why equals two for uh 4 to the Power four X to the Power one x 4. Right? So this is so if we subtract this we will get this reason right, isn't it? So this is very easy. This becomes very easy here. Now we will see how to do so using washer method. This is called washing method in another. Thank you. So This is rotated around by equals to zero night. Remember about by a question jake. So now we can say volume will reward one. We equals two. Bye. Okay by equals to A. And that is B. Yeah, this is r squared minus I but it gets yeah. So here X equals to a Right equals to zero because we are We are having this limit. Right? So this limit is what this limit is 02 when you can see. So this is why equals to be Right. So y equals two. B is equal to that is you can say the limits need global immediately abandon it. You can say so in this way we will solve this. Now we becomes word. Now as I have already told that our square is four here, why equals to four. We will take Thanks. So we will right here like this. Bye. It should be mhm X equals to zero date. So here that this will be one And this will be 44 square This whole area with four. My question for discovered. So 4 -4 squared minus four eggs -1ra. The public one x 4 here expect So I think you have got why we're doing this because from the whole area we are just subtracting this unwanted area because we want our two. Right? So no we becomes here after integration we will get bye 16 x minus 16 X. You can say when one by two plus one that is today by do divided right? Yeah. T B do only. Right. So here it will be limits. Exit question zero. This is what? Okay so this is very simple because we know that this formula is integration of facts to the power. Nd exist nothing but X to the power and plus one by N plus one plus C. That is our constant. This is very simple. So now where they're moving we will just put the limit we will get bye 16 -325 today. Yeah. Right. And it will be minus the minus lower limit here. Mhm No it becomes life B. E question. Yeah. Bye pressure. But the aid -32 divided by today. That would be 16. Bye bye today here. Now we can easily say that this is the volume covered by the rotation about by Costa zero. You can say Yes The volume is 16. This will be 16. Bye bye three. Yeah. So here 16 Fiber three is in cuba Kunitz straight. Yeah. So it can also be Simplified with this. It becomes 16.755. Kill big your notes. So this is a answer to this problem. I hope you understood the concept. Thank you for watching.