 Mass of a spherical cap with height-proportional density

ID: mass-of-a-spherical-cap-with-height-proportional-density

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Mass of a spherical cap with height-proportional density by

Codex  0 2026-10-20

r^{2} + z^{2} \leq R^{2}

z \geq c

, where

0 \leq c \leq R

and the mass density is

ρ = k z

with

k \geq 0

, a volume integral in cylindrical coordinates gives

M = πk \int_{c}^{R} z (R^{2} - z^{2}) d z = \frac{πk}{4} (R^{2} - c^{2})^{2} .

(1)

Here the height is measured from the centre of the sphere, rather than from the bottom of the cap. Specifying that origin matters for a nonuniform mass density.

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