Page 92 - Vector Analysis

88 CHAPTER 3. Multiple Integrals

Example 3.30. In this example we compute the volume ωn of the n-dimensional unit ball.

By the Fubini theorem,

??
ż 1 ż 1´x21 ż 1´x12´¨¨¨´x2n´1

ωn = ? ¨¨¨ ? dxn ¨ ¨ ¨ dx1 .

´1 ´ 1´x12 ´ 1´x21´¨¨¨´xn2´1

??
1´x21 1´x12´¨¨¨´xn2´1
ż ż n´1
? ¨¨¨ ? 2
Note that the integral dxn ¨ ¨¨ dx2 is in fact ωn´1(1 ´ x12 ) , the

´ 1´x21 ´ 1´x21´¨¨¨´xn2´1

volume of (n ´ 1)-dimensional ball of radius a1 ´ x12; thus

ż1 ż π
ωn = 2
x2 n´1 cosn (3.8)
ωn´1(1 ´ ) 2 dx = 2 ωn´1 θ dθ .

´1 0

Integrating by parts,

ż π ż π ˇθ= π ż π
2 2 θˇ 2 1) 2
cosn cosn´1 d(sin θ) cosn´1 sin (n cosn´2 θ sin2
θ dθ = θ = θ ˇθ=0 + ´ θ dθ

00 0

ż π
1)
(n 2 cosn´2 θ(1 ´ cos2

= ´ θ) dθ

0

which implies that żπ cosn θ dθ = n ´ 1 żπ cosn´2 θ dθ .
As a consequence, 2 n 2
0 0

$ (n ´ 1)(n ´ 3) ¨ ¨ ¨ 2 ż π
2
cos θ dθ if n is odd ,
ż π ’ n(n ´ 2) ¨ ¨ ¨ 3 0 if n is even ,
2 ’
cosn &

θ dθ = (n ´ 1)(n ´ 3) ¨ ¨ ¨ 1 π
n(n ´ 2) ¨ ¨ ¨ 2
0’ ż2 dθ
’
% 0

and the recursive formula (3.8) implies that ωn = 2 ωn´2 π . Further computations shows

that n

$ (2π) n´1 if n is odd ,
2
’ n(n ´ 2) ¨ ¨ ¨ 3 ω1
’
’
&

ωn = n´2

’ (2π) 2 ¨ ¨ 4 ω2 if n is even .
’ n(n ´ 2) ¨
’
%

ż8

Let Γ be the Gamma function defined by Γ(t) = xt´1e´x dx for t ą 0. Then Γ(x + 1) =

xΓ(x) for all x ą 0, Γ(1) = 1 and ( 1 ) = ?π. 0 fact that ω1 = 2 and ω2 = π, we can
Γ
2 By the

express ωn as ωn = Γ(πn+n22 2 ) .