Multivariate Calculus - Bruce E. Shapiro

LECTURE 20. TRIPLE INTEGRALS 167
Triple Integrals in Spherical Coordinates
Recall that each point in space is represented by a triple (ρ, θ, φ) where
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
and
ρ 2 = x 2 + y 2 + z 2
tan θ = y/z
cos φ = z/ √ x 2 + y 2 + z 2
The volume element in spherical coordinates is
and hence the triple integral is
V f(x, y, z)dV =
dV = ρ 2 sin φdρdθdφ
s f(ρ sin φcosθ, ρ sin φ sin θ, ρ cos φ)ρ2 sin φ dρ dθ dφ
The total volume of V is thus
V =
s ρ2 sin φdρdθdφ
Example 20.5 Find the volume of ball of radius a centered at the origin.
Solution.
V =
=
s ρ2 sin φdρdθdφ
∫ π ∫ 2π ∫ a
0 0
= 1 3 a3 ∫ π
0
= 1 3 a3 (2π)
0
∫ 2π
0
∫ π
0
ρ 2 sin φdρdθdφ
dθ sin φdφ
sin φdφ
= 2π 3 a3 (− cos φ)| π 0
= 2π 3 a3 (−(−1) − −(1)) = 4π 3 a3
Math 250, Fall 2006 Revised December 6, 2006.