Total Mass in a Cube vs. a Sphere | Proof and Derivation

The problem required computing the total mass contained within a cube and a sphere, both defined by a characteristic length $L$, subject to a quadratic density distribution $\rho(x)=\frac{\rho_0}{L^2} x^2$ (density increases quadratically with distance from the origin). By integrating the density over the respective volumes, the total mass in the cube was found to be $M_{\text {cube }}=\rho_0 L^3$. Converting to spherical coordinates was necessary for the sphere, where $d V=r^2 \sin \theta d r d \theta d \phi$, resulting in a total mass of $M_{\text {sphere }}=\frac{4}{5} \pi \rho_0 L^3$. The key takeaway is that the spherical volume, having a total mass approximately 2.51 times greater than the cube, efficiently captures the high-density regions far from the origin due to its geometry, despite having a smaller overall volume $\left(\frac{4}{3} \pi L^3 \approx 4.189 L^3\right)$ compared to the cube's volume $\left(L^3\right)$.

🎬Narrated Video

https://youtu.be/zyZt4SlRYhI

🏗️Structural clarification of Poof and Derivation

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