---
title: "Volume of sphere by triple integration"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:volume-sphere-triple-integration"
source: https://www.bctengineer.com/q/semester_2/mathematics/volume-sphere-triple-integration
site: BCT Engineering
---

# Volume of sphere by triple integration

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2073 Bhadra, 5 marks
- Video: [1.3.4 Triple Integration Engineering Maths 2: Finding the Volume of a Sphere](https://www.youtube.com/watch?v=fx1mVPH3_iI) — Apna Tuition
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/volume-sphere-triple-integration
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Find by triple integration the volume of the sphere x²+y²+z² = a².

## Short answer

By symmetry, V = 8 times the first-octant volume. Integrate z from 0 to √(a²−x²−y²), y from 0 to √(a²−x²), x from 0 to a, giving V = (4/3)πa³.

## Answer

**Method.** By symmetry, V = 8 × (volume in the first octant), where 0 ≤ z ≤ √(a² − x² − y²), 0 ≤ y ≤ √(a² − x²), 0 ≤ x ≤ a.

```text
V = 8 ∫₀^a ∫₀^{√(a²−x²)} ∫₀^{√(a²−x²−y²)} dz dy dx
```

#### z-integration
∫₀^{√(a²−x²−y²)} dz = √(a² − x² − y²).

#### y-integration
Put k² = a² − x² (constant for this step). Then

```text
∫₀^k √(k² − y²) dy = [ (y/2)√(k²−y²) + (k²/2) sin⁻¹(y/k) ]₀^k
                   = (k²/2)(π/2) = πk²/4 = π(a² − x²)/4
```

#### x-integration
```text
V = 8 ∫₀^a (π/4)(a² − x²) dx
  = 2π [ a²x − x³/3 ]₀^a
  = 2π (a³ − a³/3)
  = 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. V is computed as:
   A. first octant only
   B. 8 × first octant
   C. 4 × first octant
   D. 2 × first octant
   Answer: B. Eight octants. (Key point: Symmetry)

2. ∫₀^k √(k²−y²) dy equals:
   A. k²/2
   B. πk/4
   C. πk²/4
   D. πk²/2
   Answer: C. Quarter circle area. (Key point: y-integral)

3. V =
   A. (2/3)πa³
   B. (4/3)πa³
   C. πa³
   D. 4πa²
   Answer: B. 2π(2a³/3). (Key point: Result)

## Related in Mathematics

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- [Volume of sphere using Dirichlet's integral](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Avolume-sphere-using-dirichlet-s) — Mathematics, chapter 2

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