how to calculate mass in a non-uniform density field by using volume integration

The two density distribution demonstrations showcase how mass accumulates in various 3D geometries under a spherically dependent quadratic density function $\left(\rho(x) \propto r^2\right)$. The first visualization compared a cube and a sphere, revealing that the sphere contains significantly more mass relative to its volume because its uniform boundary maximizes the capture of high-density material far from the origin. The second, more complex demo extended this principle to an ellipsoid and a torus, effectively illustrating that mass is concentrated along the parts of the objects farthest from the coordinate center (e.g., the outer shell of the ellipsoid and the large outer rim of the torus), visually confirming that the total mass calculations are dominated by the regions where the radial distance $r$ is maximized.

https://youtu.be/zyZt4SlRYhI

🪜State Diagram: Visualizing Variable Density Fields Across Complex Geometries

The state diagram follows a progression from fundamental volume integration to the visualization of density across complex geometries.

stateDiagram-v2
    [*] --> Demo_1: Introduction to Volume Integration
    
    state Demo_1 {
        Cube_Mass --> Sphere_Mass: Cycle to compare distinct total mass values
        Sphere_Mass --> Cube_Mass: Cycle to reinforce analytical derivation
    }
    
    Demo_1 --> Demo_2: Transition to Spatial Distribution Analysis
    
    state Demo_2 {
        Side_by_Side_Comparison: Cube & Sphere displayed simultaneously
    }
    
    Demo_2 --> Demo_3: Extension to Complex Manifolds
    
    state Demo_3 {
        Ellipsoid_vs_Torus: Quadratic density in intricate shapes
    }
    
    Demo_3 --> [*]: Final visualization of non-uniform mass fields

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