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.
The state diagram illustrates the logical progression from the initial derivation of a hemisphere to the generalized calculation of a spherical cap, including the accompanying animations for both.
stateDiagram-v2
[*] --> Example1 : Hemisphere Analysis
state Example1 {
Area_Derivation --> Volume_Derivation
}
Example1 --> Demo1 : Visualize volume accumulation
state Demo1 {
direction lr
Radial_Sweep --> Volume_Fill
}
Demo1 --> Example2 : Generalize to partial sphere (z >= h)
state Example2 {
Adjust_Limits --> Cap_Volume_Formula
}
Example2 --> Demo2 : Visualize constrained integration
state Demo2 {
direction lr
Constrained_Sweep --> Cap_Visualization
}
Demo2 --> [*] : Completion
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