Hints and answers

6.6 HINTS AND ANSWERS
(a) Let R be a real positive number and define Km by
R
2
m
Km =
R − x2 dx.
−R
Show, using integration by parts, that Km satisfies the recurrence relation
(2m + 1)Km = 2mR 2 Km−1 .
(b) For integer n, define In = Kn and Jn = Kn+1/2 . Evaluate I0 and J0 directly
and hence prove that
In =
22n+1 (n!)2 R 2n+1
(2n + 1)!
and
Jn =
π(2n + 1)!R 2n+2
.
22n+1 n!(n + 1)!
(c) A sequence of functions Vn (R) is defined by
V0 (R) = 1,
Vn (R) =
R
−R
Vn−1
√
R 2 − x2 dx,
n ≥ 1.
Prove by induction that
V2n (R) =
π n R 2n
,
n!
V2n+1 (R) =
π n 22n+1 n!R 2n+1
.
(2n + 1)!
(d) For interest,
(i) show that V2n+2 (1) < V2n (1) and V2n+1 (1) < V2n−1 (1) for all n ≥ 3;
(ii) hence, by explicitly writing out Vk (R) for 1 ≤ k ≤ 8 (say), show that the
‘volume’ of the totally symmetric solid of unit radius is a maximum in
five dimensions.
6.6 Hints and answers
6.1
6.3
6.5
6.7
6.9
6.11
6.13
6.15
6.17
6.19
6.21
6.23
√
√
For integration order z, y, x, the limits are (0, √
a − x),√(− 4ax, 4ax) and (0, a).
For integration order y, x, z, the limits are (− 4ax, 4ax), (0, a − z) and (0, a).
V = 16a3 /15.
1/360.
(a) Evaluate 2b[1 − (x/a)2 ]1/2 dx by setting x = a cos φ;
(b) dV = π × a[1 − (z/c)2 ]1/2 × b[1 − (z/c)2 ]1/2 dz.
Write sin3 θ as (1 − cos2 θ) sin θ when integrating |Ψ2 |2 .
(a) V = 2πc × πa2 and A = 2πa × 2πc. Setting ro = c + a and ri = c − a gives the
stated results. (b) Show that the centre of gravity of either half is 2a/π from the
cylinder.
Transform to cylindrical polar coordinates.
4πa2 ; 4πa3 /3; a sphere.
The volume element is ρ dφ dρ dz. The integrand for the final z-integration is
given by 2π[(z 2 ln z) − (z 2 /2)]; I = −5π/9.
Set ξ = x/a, η = y/b, ζ = z/c to map the ellipsoid onto the unit sphere, and then
change from (ξ, η, ζ) coordinates to spherical polar coordinates; I = 4πa3 bc/15.
Set u = sinh x cos y and v = cosh x sin y; Jxy,uv = (sinh2 x + cos2 y)−1 and the
integrand reduces to 4uv over the region 0 ≤ u ≤ 1, 0 ≤ v ≤ 1; I = 1.
Terms such as T ∂2 S/∂Y ∂X cancel in pairs. Use equations (6.17) and (6.16).
(c) Show that the two expressions mutually support the integration formula given
for computing a volume in the next higher dimension.
(d)(ii) 2, π, 4π/3, π 2 /2, 8π 2 /15, π 3 /6, 16π 3 /105, π 4 /24.
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