Volume of a Sphere using Spherical Coordinates

Example: Find the volume, V, of a sphere of radius, R, using spherical coordinates.

First integration on the variable ρ.

θ = 2π φ = π ρ = R

V = ∫ ∫ ∫ ρ² sin φ dρ dφ dθ

θ = 0 φ = 0 ρ = 0

θ = 2π φ = π R

V = ∫ ∫ ρ³ / 3| sin φ dφ dθ

θ = 0 φ = 0 0

Second integration on the variable φ .

θ = 2π φ = π θ = 2π π

V = ∫ ∫ (R³ / 3) sin φ dφ dθ = (R³ / 3) ∫ - cos φ | dθ

θ = 0 φ = 0 θ = 0 0

Third integration on the variable θ .

θ = 2π

V = ∫ 2 R³ / 3 dθ = 4 π R³ / 3 (result)

θ = 0

Notice the ease of determining the limits of integration when using spherical coordinates

when compared to the use of rectangular coordinates to evaluate the integral.

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