Comments on: Make my math dreams come true
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Comments on Ask MetaFilter post Make my math dreams come trueMon, 14 Mar 2022 16:40:47 -0800Mon, 14 Mar 2022 17:15:38 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: Make my math dreams come true
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true
How to derive the volume of a sphere from the area of a circle, integrating along rotational angle. <br /><br /> I had a crazy intense dream last night where I was trying to derive the area of a sphere by knowing that the area of a circle is pi*r^2 and then having it rotate about an axis from 0 to 360 and integrating along rotational angle. Haven't found anything on YouTube nor by trying to work it out via university calculus that is 20 years dusty.<br>
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Not the cylinder method nor the pyramid method I don't think. Is this even possible?<br>
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Thanks in advance it's been bugging me all day.post:ask.metafilter.com,2022:site.361768Mon, 14 Mar 2022 16:40:47 -0800St. PeepsburgPispherevolumecalculusresolvedBy: Ryon
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159725
This math is a little beyond me, but I love a good Google challenge. Does this help you at all?<br>
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<a href="https://www.math.stonybrook.edu/~mde/141F_02/gw4.pdf">Compute the volume of a sphere of radius r using an integral</a> (PDF link)comment:ask.metafilter.com,2022:site.361768-5159725Mon, 14 Mar 2022 17:15:38 -0800RyonBy: alexei
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159728
If you're integrating from 0 to 360, I think you'd have to be integrating a wedge. You can calculate the volume of a wedge of angular thickness dϕ and radius R as the integral from z = 0 to R of (z⋅dϕ)⋅(dz)⋅(2⋅√(R²-z²)), which is 2/3⋅R³⋅dϕ. Then easily integrate that from ϕ = 0 to 2π which gets you 4/3⋅π⋅R³.comment:ask.metafilter.com,2022:site.361768-5159728Mon, 14 Mar 2022 17:24:52 -0800alexeiBy: pmdboi
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159736
A sphere is a solid of revolution where the shape is a semicircle being revolved about its flat edge. So you could also use <a href="https://math24.net/pappus-theorem.html">Pappus's second centroid theorem</a>, which says that the volume of a solid of revolution is equal to the product of the area of the shape being revolved and the distance traveled by that shape's <a href="https://tutorial.math.lamar.edu/classes/calcii/centerofmass.aspx">centroid (AKA center of mass)</a>.<br>
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The <a href="https://datagenetics.com/blog/january52017/index.html">centroid of a semicircle of radius <i>r</i></a> lies 4<i>r</i>/3π from the edge, so the distance the centroid travels is 4<i>r</i>/3π × 2π = 8<i>r</i>/3. The area of the semicircle is π<i>r</i><sup>2</sup>/2. So the volume of the sphere is the product 8<i>r</i>/3 × π<i>r</i><sup>2</sup>/2 = 4π<i>r</i><sup>3</sup>/3.comment:ask.metafilter.com,2022:site.361768-5159736Mon, 14 Mar 2022 17:48:18 -0800pmdboiBy: zeptoweasel
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159744
That sounds like a cool dream! You can totally compute the volume of a sphere by figuring out the volume of an infinitesimal wedge and rotating that around an axis.<br>
1. Imagine a semicircle with radius R sitting on the x axis, in the x-y plane. Now rotate it a little bit out of the page to make a thin wedge. <br>
2. Consider a vertical slice of the wedge whose width in the x direction is dx. The volume of this slice is dx times the area of a triangle with height y=sqrt(R<sup>2</sup>-x<sup>2</sup>) and base y dθ=sqrt(R<sup>2</sup>-x<sup>2</sup>)dθ. This works out to (R<sup>2</sup>-x<sup>2</sup>)/2 dx dθ.<br>
3. Now integrate this expression over x from -R to R to get the volume of the wedge. It's a nice simple polynomial in x, which is convenient. We get (2/3)R<sup>3</sup> dθ.<br>
4. And finally rotate the wedge! That is, integrate over θ from 0 to 2π. The integrand doesn't depend on θ, so we're just multiplying by 2π. Result: the volume of a sphere is (4/3)πR<sup>3</sup>.comment:ask.metafilter.com,2022:site.361768-5159744Mon, 14 Mar 2022 18:13:31 -0800zeptoweaselBy: St. Peepsburg
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159762
Thanks everyone! All of these answers are great, I marked the ones that were most step by step / used angles and integrals similar to my dream / easy to understand after forgetting all my trig and calculus. You da best!comment:ask.metafilter.com,2022:site.361768-5159762Mon, 14 Mar 2022 19:20:50 -0800St. PeepsburgBy: platinum
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159799
A timely dream for Pi Day!comment:ask.metafilter.com,2022:site.361768-5159799Mon, 14 Mar 2022 22:31:24 -0800platinumBy: James Scott-Brown
http://ask.metafilter.com/361768/Make-my-math-dreams-come-true#5159873
The first few pages of <a href="https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-volumes-2009-1.pdf">this document</a> addresses this.comment:ask.metafilter.com,2022:site.361768-5159873Tue, 15 Mar 2022 05:50:50 -0800James Scott-Brown