---
title: "Volume of sphere using Dirichlet's integral"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:volume-sphere-using-dirichlet-s"
source: https://www.bctengineer.com/q/semester_2/mathematics/volume-sphere-using-dirichlet-s
site: BCT Engineering
---

# Volume of sphere using Dirichlet's integral

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2069 Poush, 5 marks
- Video: [L-39.2 Volume Integration  Using Dirichlet  Integral](https://www.youtube.com/watch?v=iZWVMB-w5OI) — TECH-MATH
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/volume-sphere-using-dirichlet-s
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

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

## Short answer

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

## Answer

**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).

#### Apply to the sphere
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.

```text
∭ 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:

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

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

#### Total
```text
V = 8 · π a³/6 = 4π a³/3
```

**Answer: V = (4/3)π a³.**

## Check yourself

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

1. Why is the volume found for the first octant and multiplied by 8?
   A. Dirichlet's integral needs x, y, z >= 0 and the sphere is symmetric about all three coordinate planes
   B. The sphere has 8 equal radii
   C. Dirichlet's integral only works for a cube
   D. Because the integrand is 8 times larger there
   Answer: A. The formula holds for x, y, z >= 0, and symmetry makes the 8 octants equal. (Key point: Why use the first octant)

2. Which values of p, q, r and l, m, n are used for the volume of the sphere?
   A. p = q = r = 1 and l = m = n = 2
   B. p = q = r = 2 and l = m = n = 2
   C. p = q = r = 2 and l = m = n = 1
   D. p = q = r = 3 and l = m = n = 1
   Answer: C. The surface is (x/a)^2 + (y/a)^2 + (z/a)^2 = 1, so p = q = r = 2. The integrand is 1, so each exponent l - 1 = 0, giving l = m = n = 1. (Key point: Choosing the parameters)

3. What is Gamma(5/2) in terms of sqrt(pi)?
   A. (1/2) sqrt(pi)
   B. (3/4) sqrt(pi)
   C. (3/2) sqrt(pi)
   D. sqrt(pi)
   Answer: B. Gamma(5/2) = (3/2)(1/2)Gamma(1/2) = (3/4) sqrt(pi). (Key point: Gamma values)

4. What is the first-octant volume, and hence the total volume?
   A. pi a^3/6, so V = 4 pi a^3/3
   B. pi a^3/8, so V = pi a^3
   C. pi a^3/3, so V = 8 pi a^3/3
   D. 4 pi a^3/3, so V = 32 pi a^3/3
   Answer: A. (a^3/8)(4 pi/3) = pi a^3/6, and 8 times this is 4 pi a^3/3. (Key point: First octant volume and total)

## Related in Mathematics

- [Volume of sphere by triple integration](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Avolume-sphere-triple-integration) — Mathematics, chapter 2
- [Volume common to two cylinders by triple integral](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Avolume-common-two-cylinders-triple) — Mathematics, chapter 2
- [Dirichlet integral over positive octant region](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adirichlet-integral-over-positive-octant) — Mathematics, chapter 2, asked 3×

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