In considering volumes created by revolving polynomials $y=\beta x^n$ about the y-axis, if we specify $\beta$ so that the curve includes $(0,0)$ and $(a,2a)$ and consider the volumes swept within the cylinder of radius $a$ and height $2a$ defined between $x=[-a,a]$, I was mildly surprised to realize that several simple polynomials sweep out volumes equivalent to a sphere ($2/3$ the cylinder volume). In general, the partition is a bullet shape above the curve, a cup shape below, with fractional partitions of the cylinder volume of $[\frac{n}{n+2},\frac{2}{n+2}]$. These volumes can be reformulated symmetrically about $y=a$ to form central solids versus their surrounding cavities.
As an example, $y=\frac{2}{a^3}x^4$ sweeps a bullet with $2/3$ the cylinder volume, the same as a sphere, and this can be made into a symmetric UFO-like shape. Or, we could take the paraboloid bullet from $n=2$ with volume $1/2$ the cylinder and add to it the cup from $n=10$ with volume $1/6$ the cylinder to obtain $2/3$. There are many such combinations.
I'm familiar with some bottom-up geometric constructs to derive the formula for a sphere. Archimedes and Cavalieri used cylinders with displaced cones (cylinder is $n=0$ above with cup volume $1$, cone is $n=1$ above with bullet volume $1/3$) in their constructs. I've also seen a Chinese proof involving a bicylinder within a cube. All these use some equality or proportionality between moving sections of the figures. I've (unsurprisingly, given how clever the proofs are) been unable to see any such application of the revolved quadratic, quartic, or other polynomials above, other than the cone.
My question: Are there geometric constructs to derive formulas for spheres that leverage one or more of these polynomial volume combinations, beyond those I've mentioned above? Or is it known that there are none? Thanks in advance!