---
title: "Dirichlet integral over positive octant region"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:dirichlet-integral-over-positive-octant"
source: https://www.bctengineer.com/q/semester_2/mathematics/dirichlet-integral-over-positive-octant
site: BCT Engineering
---

# Dirichlet integral over positive octant region

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2072 Magh, 5 marks; 2070 Magh, 5 marks; 2067 Mangsir, 5 marks (asked 3 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/dirichlet-integral-over-positive-octant
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate: ∭ x^(l−1) y^(m−1) z^(n−1) dx dy dz, where x, y, z are all positive but (x/a)^p + (y/b)^q + (z/c)^r ≤ 1.

## Short answer

Substitute u=(x/a)^p, v=(y/b)^q, w=(z/c)^r; region becomes u+v+w≤1. By Dirichlet’s theorem the integral equals (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).

## Answer

This is Dirichlet's integral for the generalised region (x/a)^p + (y/b)^q + (z/c)^r ≤ 1 with x, y, z ≥ 0.

#### Dirichlet's standard result
For u, v, w ≥ 0 with u + v + w ≤ 1:

∭ u^(α−1) v^(β−1) w^(γ−1) du dv dw = Γ(α) Γ(β) Γ(γ) / Γ(α + β + γ + 1).

#### Substitution
Put (x/a)^p = u, (y/b)^q = v, (z/c)^r = w. Then the region becomes u + v + w ≤ 1, and

```text
x = a u^(1/p)   →  dx = (a/p) u^(1/p − 1) du
y = b v^(1/q)   →  dy = (b/q) v^(1/q − 1) dv
z = c w^(1/r)   →  dz = (c/r) w^(1/r − 1) dw
```

So x^(l−1) dx = a^(l−1) u^((l−1)/p) · (a/p) u^(1/p − 1) du = (a^l/p) u^(l/p − 1) du. Similarly for y and z.

#### The integral
```text
I = (a^l b^m c^n)/(p q r) ∭ u^(l/p − 1) v^(m/q − 1) w^(n/r − 1) du dv dw
```

Applying Dirichlet's result with α = l/p, β = m/q, γ = n/r:

```text
I = (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)
```

**Special case check.** For l = m = n = p = q = r = 1 and a = b = c = 1 (the tetrahedron x + y + z ≤ 1): I = 1/Γ(4) = 1/6, which is the correct volume.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which substitution reduces the region to u+v+w ≤ 1?
   A. (x/a)^p = u, (y/b)^q = v, (z/c)^r = w
   B. x = u, y = v, z = w
   C. x = a u^p etc.
   D. u = x/a + y/b + z/c
   Answer: A. This maps the region to the standard simplex. (Key point: Substitution)

2. The Dirichlet parameters are:
   A. l/p, m/q, n/r
   B. l, m, n
   C. lp, mq, nr
   D. p/l, q/m, r/n
   Answer: A. x^(l−1)dx = (a^l/p) u^(l/p − 1) du. (Key point: Parameters)

3. The final formula is:
   A. Γ(l)Γ(m)Γ(n)/Γ(l+m+n+1)
   B. a^l b^m c^n Γ(l)Γ(m)Γ(n)/Γ(l+m+n)
   C. (a^l b^m c^n)/(pqr) · Γ(l/p)Γ(m/q)Γ(n/r)/Γ(l/p+m/q+n/r+1)
   D. (pqr)/(a^l b^m c^n) · Γ(l/p)Γ(m/q)Γ(n/r)/Γ(l/p+m/q+n/r)
   Answer: C. Includes the constant factor a^l b^m c^n/(pqr) and +1 in the denominator. (Key point: Result)

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