Why is the volume found for the first octant and multiplied by 8?

Dirichlet's integral needs x, y, z >= 0 and the sphere is symmetric about all three coordinate planes

Dirichlet's integral only works for a cube

Because the integrand is 8 times larger there

Which values of p, q, r and l, m, n are used for the volume of the sphere?

p = q = r = 2 and l = m = n = 1

p = q = r = 1 and l = m = n = 2

p = q = r = 2 and l = m = n = 2

p = q = r = 3 and l = m = n = 1

Short answer

Using Dirichlet's integral for the first octant gives volume πa³/6. Multiplying by 8 yields the sphere volume V = 4πa³/3.

Find the volume of sphere x²+y²+z² = a² using Dirichlet's integral.

Dirichlet's integral. For x, y, z ≥ 0 with (x/a)^p + (y/b)^q + (z/c)^r ≤ 1,

∭ x^(l−1) y^(m−1) z^(n−1) dx dy dz = (a^l b^m c^n)/(p q r) · Γ(l/p) Γ(m/q) Γ(n/r) / Γ(l/p + m/q + n/r + 1).

The sphere x² + y² + z² = a² is symmetric about all three coordinate planes, so

V = 8 × (volume in the first octant) = 8 ∭ dx dy dz over x² + y² + z² ≤ a², x, y, z ≥ 0.

Write the region as (x/a)² + (y/a)² + (z/a)² ≤ 1, so a = b = c, p = q = r = 2. For the volume, the integrand is 1, so l = m = n = 1.

∭ dx dy dz = (a·a·a)/(2·2·2) · Γ(1/2)³ / Γ(3/2 + 1)
           = (a³/8) · Γ(1/2)³ / Γ(5/2)

Use Γ(1/2) = √π and Γ(5/2) = (3/2)(1/2)Γ(1/2) = (3/4)√π:

Γ(1/2)³ / Γ(5/2) = π√π / ((3/4)√π) = 4π/3

first-octant volume = (a³/8)(4π/3) = π a³/6

V = 8 · π a³/6 = 4π a³/3

Answer: V = (4/3)π a³.

✓ Answer checked by Claude Opus 5.5 (high effort)