🎬Volume Visualization of Spherical Geometry-Hemisphere and Cap Integration using Cylindrical Coordina

The two visualizations serve as animated demonstrations of volume integration in cylindrical coordinates $(\rho, \phi, z)$ for spherical geometry. The first demonstration calculated the volume of a full hemisphere ( $\frac{2}{3} \pi R^3$ ) by integrating the volume element $\rho d \rho d \phi d z$ over a radial distance from 0 to $R$, showing the accumulation of full-height cylindrical discs. The second demonstration generalized this method to a spherical cap sliced by a plane at height $h$. It visualized the accumulation of volume as the radial coordinate $\rho$ swept from the pole (0) to the cap's base radius $\left(\sqrt{R^2-h^2}\right)$, with the vertical limits of integration dynamically constrained between the fixed slicing plane $z=h$ and the sphere's curved surface $z=\sqrt{R^2-\rho^2}$, confirming the derived volume formula for the cap.

🎬Narrated Video

🧵Related Derivation

🧄Calculating the Area of a Half-Sphere Using Cylindrical Coordinates (AHS-CC)

⚒️Compound Page

Volume Visualization of Spherical Geometry-Hemisphere and Cap Integration using Cylindrical Coordinates | Resulmationviadean on Notion