Answer:
V = π (rh² − ⅓h³)
Explanation:
Slice the volume horizontally into a stack of disks.
Each disk has a radius R at a position y from the center, and a thickness of dy.
The volume of each disk is:
dV = π R² dy
Using Pythagorean theorem:
R² + y² = r²
R² = r² − y²
Substituting:
dV = π (r² − y²) dy
Integrate from y = r−h to y = r.
V = ∫ dV
V = ∫ π (r² − y²) dy
V = π (r²y − ⅓y³)
Evaluating between the limits:
V = π [r²(r) − ⅓r³] − π [r²(r−h) − ⅓(r−h)³]
V = π (⅔r³) − π [r³ − r²h − ⅓(r³ − 3r²h + 3rh² − h³)]
V = π (⅔r³) − π (r³ − r²h − ⅓r³ + r²h − rh² + ⅓h³)
V = π (⅔r³) − π (⅔r³ − rh² + ⅓h³)
V = π (rh² − ⅓h³)
Check the answer.
If h = 0, V = 0, as expected.
If h = r, V = ⅔πr³, or half a sphere.