Question

Find the volume V of the described solid S.
a cap with height h of a sphere with radius r

Answer 1

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.