( a x ) 2 . 5 + ( b y ) 2 . 5 = 1 .
The poet and scientist Piet Hein created an egg that will stand on its oblong edge. He called it the superegg, which is a solid of revolution (about the y-axis) characterized by the equationCalculate the volume of the superegg if a = 2 and b = 3 .
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So I wasn't wrong the first time (except you're right, when revolved around the y-axis it looks like a curling stone). Somebody reported that the egg was not a solid of revolution, so I changed it to a general superellipsoid. I think I will create a problem called "The Curling Stone" and have the answer be 84.8262.
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Thanks for clarifying. I've updated the answer accordingly.
Yeah, Stephen, even I can be wrong sometimes. It's a very confusing subject, but, hey, it pushed me to spend time looking more closely at this and I've learned a few things.
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Since you are the only person who answered, it is not too late to change the equation for this problem (and ask the moderators to change the answer).
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@Steven Zheng – Sure, but I wonder why nobody else tried. I thought it was a nice problem.
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@Michael Mendrin – Did you use the Dirichlet Integral for the superellipsoid? I had to learn that to re-solve this problem.
Also, I think I did not type the correct axis of revolution. It's been a while since I solved such problems. I think I wrote revolve around the x-axis, while the correct axis is y. So the answer is 56.5508 despite a =2 and b=3.
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@Steven Zheng – No, I said I couldn't do it, in the case of the superellipsoid. Or at least I didn't try very hard, after I consulted Wikipedia on exactly what is a superellipsoid, and how it's different from a "superellipse". The Wikipedia article provide the formula for the volume, and I was struck by the similarity between the two, even though I couldn't get anywhere with the former.
Excuse me. How did you integrated the function? Can you give me a reference to the method? Thanks!
We can calculate the volume on the side x > 0 then multiply with two (by symmetry).
For x ≥ 0 the equation reduces to : y = f ( x ) = 3 ( 1 − ( 2 x ) 5 / 2 ) 2 / 5 We know that : V = 4 π ∫ 0 2 x ∣ f ( x ) ∣ d x = x = 2 t 8 π ∫ 0 1 t f ( 2 t ) d t = 4 8 π ∫ 0 1 t ( 1 − t 5 / 2 ) 2 / 5 d t . Now set z = t 5 / 2 then t d t = 5 2 z − 1 / 5 , and then : V = 5 9 6 π ∫ 0 1 z − 1 / 5 ( 1 − z ) 2 / 5 d z = 5 9 6 π B ( 5 4 , 5 7 ) Where B is the Beta function , use the Beta-Gamma relation and then Γ ( x + 1 ) = x Γ ( x ) to get : V = 5 9 6 π ⋅ Γ ( 5 1 1 ) Γ ( 5 7 ) Γ ( 5 4 ) = 3 2 π Γ ( 5 1 ) Γ ( 5 4 ) Γ ( 5 2 ) Use WolframAlpha to get that the answer is ≈ 5 6 . 5 5 0 8 .
Remark 1: We can use the reflection formula to get V = sin 5 π 3 2 π 2 ⋅ Γ 2 ( 5 1 ) Γ ( 5 2 ) You can even use python but you need more work.
Remark 2: Using the same method we can generalize the result like what @Michael Mendrin has found.
So that's how it's done, in the case of the superellipisoid.
That's how I did it too (including the W|A for evaluating Gamma functions).
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The volume formula for Piet Hein's superellipse (not to be confused with superellipsoid) is
2 π a b 2 Γ ( 1 + n 3 ) Γ ( 1 + n 1 ) Γ ( 1 + n 2 )
so that for n = 2 5 , a = 3 , and b = 2 (as suggested by the photograph of Hein's Egg), the volume works out to 5 6 . 5 5 0 8 . . . . For a = 2 and b = 3 , it works out to 8 4 . 8 2 6 2 . . . .
The volume of the superellipsoid, however, is
8 a b c Γ ( 1 + n 3 ) ( Γ ( 1 + n 1 ) ) 3
so that for n = 2 5 , a = 3 , and b = c = 2 it works out to 6 0 . 8 5 9 1 . . . .
The difference between Piet Hein's superellipse and the general superellipsoid is that the latter is not a solid of revolution.