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Problem 71 Hard Difficulty

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the
volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.
(a) A barrel with height $ h $ and maximum radius $ R $ is constructed by rotating about the x-axis the parabola $ y = R - cx^2 $, $ \frac{-h}{2} \le x \le \frac{h}{2} $, where c is a positive constant. Show that the radius of each end of the barrel is $ r = R - d $, where $ d = \frac{ch^2}{4} $.
(b) Show that the volume enclosed by the barrel is
$$ V = \frac{1}{3} \pi h (2R^2 + r^2 - \frac{2}{5} d^2) $$

Answer

a) $R-d=r$
b) $V=\frac{1}{3} \pi h\left(2 R^{2}+r^{2}-\frac{2}{5} d^{2}\right)$

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Video Transcript

Okay we're gonna send this parabola which looks like this pretend like and it's going from wine. Shh over 22 H. Over two and then send it around the X. Axis and that will give us a barrel. Okay to find the volume. We're going to cut it vertically so that we have disks and the volume of a disk has pi r squared H. And age is the thickness. And you can see that the thicknesses D. X. And R. Is the radius. So from the X. Axis up you can tell that some why value which in this case is R minus C. X. Squared. So the volume is pi integral R minus C. X. Squared square D. X. From minus H. Over 22 H. Over to. Well this is symmetric so I'm going to do to hear and zero to H. Over to Okay so now I got to pie in a girl I gotta squared out so R squared minus two R. C. X squared plus C squared X. To the fourth D. X. So two pi r squared x minus two R. C. X. cubed over three plus he squared X. To the fifth over five. From 0 to each over two push. So two pi R squared H. Over to -2 R. c over three H cubed over eight plus C squared over five H. to the 5th over 32 minus zero. Okay next thing I'm gonna do is I'm going to multiply through by two and at the same time I'm gonna factor into each out of everything. So now I have pie multiplying by two H. Uh I'm trying to be as neat as possible because it gets really gross here in a minute by over H. So there's two cancels with this too and the H. Came out so we just have our square there minus. Okay this two cancels with the eight So I have to r. c. over three H. Squared over four. I remember I took one of the H. Is out And then this to cancels with the 32 leaving 16. So I get plus 1 5th C. Squared H. to the 4th over 16. Okay so what I know is that D. Is C. H. Squared over four. That's why I wrote it in this way ch squared over four. Oops. Uh the Middle one. I'm gonna move the sea. I'd better be careful. That's our and that was ch squared over for. So now I have pie. H. R. Squared -2 are over three D. Plus 1/5 D. Squared. Okay so now I'm pretty excited about that so I look to see what happens next And I noticed I forgot to write down the answer. I'm trying to get. It was 1/3 or squared plus R. Squared minus to fifth D. Squared wow. How do I have that memorized? Oh I can't tell you how many times I worked this problem. All right 1/3 x eight in the front. Okay so I'm gonna go ahead and factor 1/3 out so that means this is now three R squared because one third times three will give you one and this is to R. D. And this is 3/5 the square. All right. So then I tried and I tried and I tried and I could not get it to turn into what it was supposed to. So here's what I did next. I said I'm just going to suppose that it is equal and then see what's the difference between what I have and what I'm trying to get. Okay? So then I said oh camera subtract big R squared from both sides. So now I have two R squared. I'm going to subtract 3/5 dared from both sides. So I'm going to move it over there. So now I have minus D squared then I still have little R squared here and I have minus two R. D. Over here. Yeah. Uh You know what that's supposed to be two R squared back up here when I'm trying to get it to be something to our square climates made a fatal error there. So now I really just have uh one or square right here. Okay, so then I said all right, I'm going to factor big are out of here and that gives me ar minus two D. And over here I'm going to factor this into our minus D. R. Plus D. Okay, well then I remembered that little are is big ar minus D. And so then R plus D. Is big are So now I have AR -2 d. Equals ar minus D. Big are so I just divide both sides by big are So now I have big ar -2 d. Equals little ar minus D. So I had to do to both sides and I get bigger as R. Plus D. Which is true, so I know that what I have here is the answer right here right here but I just got to mess with it and get it into the right form but now I know what to do, I'm just going to do what I did only backwards. Here's the one third pie age. All right and then I need the three R squared. I needed to be an R. And R squared and a two R squared. So that's the first thing I do, I'm going to keep the two R. Squared and then I'm going to make the other R squared right there. Okay so I'm never gonna mess with this two R. Squared again. All right, so now I'm gonna factor now I'm going to go back here, I'm gonna factor the big are out of the middle here, I'm not going to write the one through pi H anymore. Okay so I got the two are square plus big our times Big AR -2 D Plus 3 50 square. Alright now I'm gonna see where the Big AR -2 d. Came from. Um What Big AR -2 D. was equal to little ar minus D. Oh yeah I remember big are bigger was our little R. Plus D. So I'm going to put that in to our square plus. Wait oh yeah I'm gonna change this. Big are two little r. plus d. And this big art to little R. Plus D. Okay so now I have two R. Squared plus are plus D. And then R. Plus D. Minus two D. That's ar minus deed. Oh I'm gonna get it. Okay so I get two R. Squared plus R. Squared minus D. Squared plus 3/5 D. Squared. And that is two R. Squared plus R. Squared minus 5/5 plus 3/5 that's minus 2/5 D. Squared. Oh yeah amy. Okay so I had to I had to do this junk here which is not part of the problem. Okay so this is me working on the side, I had to do this to get an idea of how to make them equal. And the the real trick for me was first realizing that I needed to break up the three R. Squared into two R. Squared and R. Squared. So at least I got a little part of it to start with. But it was of course this middle that was the problem. So what what needed to be replaced by what? So once I realized that that's how the big ours that I could have the two R. Squared. I knew I had to get rid of this horror in this R. And then I knew what to do. Yeah. I hope you loved it.