Make my math dreams come true
March 14, 2022 4:40 PM
How to derive the volume of a sphere from the area of a circle, integrating along rotational angle.
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.
Not the cylinder method nor the pyramid method I don’t think. Is this even possible?
Thanks in advance it’s been bugging me all day.
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.
Not the cylinder method nor the pyramid method I don’t think. Is this even possible?
Thanks in advance it’s been bugging me all day.
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.
posted by alexei at 5:24 PM on March 14, 2022
posted by alexei at 5:24 PM on March 14, 2022
A sphere is a solid of revolution where the shape is a semicircle being revolved about its flat edge. So you could also use Pappus's second centroid theorem, 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 centroid (AKA center of mass).
The centroid of a semicircle of radius r lies 4r/3π from the edge, so the distance the centroid travels is 4r/3π × 2π = 8r/3. The area of the semicircle is πr2/2. So the volume of the sphere is the product 8r/3 × πr2/2 = 4πr3/3.
posted by pmdboi at 5:48 PM on March 14, 2022
The centroid of a semicircle of radius r lies 4r/3π from the edge, so the distance the centroid travels is 4r/3π × 2π = 8r/3. The area of the semicircle is πr2/2. So the volume of the sphere is the product 8r/3 × πr2/2 = 4πr3/3.
posted by pmdboi at 5:48 PM on March 14, 2022
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.
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.
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(R2-x2) and base y dθ=sqrt(R2-x2)dθ. This works out to (R2-x2)/2 dx dθ.
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)R3 dθ.
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)πR3.
posted by zeptoweasel at 6:13 PM on March 14, 2022
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.
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(R2-x2) and base y dθ=sqrt(R2-x2)dθ. This works out to (R2-x2)/2 dx dθ.
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)R3 dθ.
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)πR3.
posted by zeptoweasel at 6:13 PM on March 14, 2022
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!
posted by St. Peepsburg at 7:20 PM on March 14, 2022
posted by St. Peepsburg at 7:20 PM on March 14, 2022
The first few pages of this document addresses this.
posted by James Scott-Brown at 5:50 AM on March 15, 2022
posted by James Scott-Brown at 5:50 AM on March 15, 2022
« Older What to look for in dentists for the over-40 crowd | Amusements for the discerning senior baby/junior... Newer »
This thread is closed to new comments.
Compute the volume of a sphere of radius r using an integral (PDF link)
posted by Ryon at 5:15 PM on March 14, 2022