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The rcgion bounded by y this solid of revolution:lr? + 4c + 12,2 = and y = 0 is rotated about the y-axis_ Find thc volumc ofFind thc cxact valuc_ Write ` your answe...

Question

The rcgion bounded by y this solid of revolution:lr? + 4c + 12,2 = and y = 0 is rotated about the y-axis_ Find thc volumc ofFind thc cxact valuc_ Write ` your answer without using decimals_VolumePrcviewGet Help: VIDEOPointz pozsible: This is attempt of 3_ Message instructor about us questlon Post this question to fonumLiccnec

The rcgion bounded by y this solid of revolution: lr? + 4c + 12,2 = and y = 0 is rotated about the y-axis_ Find thc volumc of Find thc cxact valuc_ Write ` your answer without using decimals_ Volume Prcview Get Help: VIDEO Pointz pozsible: This is attempt of 3_ Message instructor about us questlon Post this question to fonum Liccnec



Answers

Find the volume of the solid of revolution formed by rotating about the $x$ -axis each region bounded by the given curves.
$$f(x)=\sqrt{4 x+2}, y=0, \quad x=0, \quad x=2$$

In this condemned of long echoed by in the government painting be f the next square D Thanks. They were given the i ego ju zero and be here echo ju ju unsound uh, function f X equals the square of the far experts do. Therefore we can ghenda Fulham and we encourage it by in car bombs. Eldridge, Jew square for experts do squared dx You're not just gonna squared and the square cancel out Then we have only for experts to the ex Im tired He lived on the inside with a coach that you x squared with you just do excellent sorry on then evaluated and resorted to doing any good to the pie Now it wouldn't you in some technical Julia juice quagga too far for them to be aid and less far on day minus zero, they get included a trail pie

Volume generated by two X plus one rotated over the X axis. So volume is equal to the integral from 0 to 4 of pi times two x plus one squared DX the pies a constant multiple. So that comes out 0242 X plus one all squared expands to four X squared plus four x plus one TX. So we're gonna take the anti derivative of each term. Four X squared would give us four over three X cubed, Ah, anti derivative of four X is gonna be two x squared, anti derivative of one his ex. And this is all being evaluated from 0 to 4. So we'll have pie. We're gonna have ah, putting four in there and will give us It's four over three times 64 plus two times 16 which is 32 plus four. And then when we put zero in there, it's good just to be subtract 000 So we don't need to do that part. When we simplify all of this, we will get the answer. 364 pi over three

Personal record on the volume. Every coach by temps in the government you b f x squared the X here were given the I hear equal judo zero be here echoed to the far on the function F X Here it is equal attuned, uh, to express one. Therefore, we can get the volume It will echo chewed up by from Joe Jafar and they will have a Jew X plus one square t x Don't get me budget up by from Jafar here The square we expanding and I'm gonna far X square Plus, uh, listen, we for express one the ANC's I'm getting coated up by now. I'm totally riveting on the inside from the X squared. Get equal to the expert 3/3. Plus first, I'm getting too. Thanks. Square on for the one getting coaching there. Thanks. Even letting others ot fall on deaf from getting coated a pie. And now here we have the far ah temps for about three hours now. Three best du temps for about you, that's far. And then we managed everything will be zero here. Then you're gonna by now. Andi, here we hand uh, yeah, we have the three in the common denominator for both far and then Plus the this one will be that you time 16 tense three, then blessed with the ground. I don't get it. Could you know 3 60 far by out of three, depending by three. So it's not nice number, so there will be 3 64 by out of three.

Finding the volume of the sort of revolution formed by rotating about X access. The region of Quebec's Zeke two X squared over two wise with 0600 back safely before is us wanting to rotate this area here about the X axis. So the shape of that will look something like this. Yeah, So it looks kind of like a cone. And remember to find this volume, it will be equal to pie integral of our bounds. So 0 to 4 and then you also have f of X, and we need to square this DX. So this will be pie 0 to 4 and then we square. So x, something's going on in here. X squared over two square defects Pi 0 to 4. So I'll be X to the fourth over four. So I'm just gonna go ahead and move that fourth out front here and then we have the DX when they go ahead and move the sex in the fourth family now integrating this will be power rule. So I have hi Force X now to the fifth power divided by five and we want to evaluate this from or 20 But smoke that up a little bit. A pi four and actually combining the four and the five here, we'll get 20. So let's go ahead and write that there instead. Yeah. Uh huh. High of 20. And now we want to evaluate this. So just X to the fifth at four and zero. So we'll have forward to the fifth power, which is 1000 24 and then plugging in zero will be zero. So 10, 24 divided by 20 Is this here will be equal to 256 over to, Uh huh. And this will be our volume.


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