I am aware you can find the center of mass of simple shapes such as those bound by parabolas using a single or double integral, but I have been trying to find a more computationally efficient method to compute this.
When you have 2 intersecting parabolas with the same magnitude of $x^2$ coefficients there seems to be a shortcut for finding the center of mass of the region they bound.
The center of mass seems to be the midpoint of the vertices of each parabola.
You can see a visual of this here: https://www.desmos.com/calculator/hr9wyik3z2
I was wondering if there was an elegant way to prove this finding, or even extend this finding for 2 intersecting parabolas with different magnitudes of $x^2$ coefficients.
If one parabola has the form $y=\alpha x^2 + bx +c$ and another one $y=-\alpha x^2+dx+e$, then we can match one parabola with another with $180^\circ$ rotation and translation. In 2D combination of rotation by some non-zero angle and translation is the same as rotation by the same angle around some other point. Thus 2 parabolas has rotational symmetry. Since after rotation the vertex of one parabola will go to the vertex of other, the center of rotation should be midpoint of the vertices. Finally, the body with rotational symmetry will always have its center of mass located in the center of rotation