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

# Volume of ellipsoid by triple integration

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2073 Magh, 5 marks
- Video: [Triple Integration |Lecture 05|Ellipsoid|Engineering Mathematics |Pradeep Gir Sir](https://www.youtube.com/watch?v=9-xpSrWrAZQ) — Pradeep Giri Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/volume-ellipsoid-triple-integration
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Find by triple integration the volume of the ellipsoid x²/a² + y²/b² + z²/c² = 1.

## Short answer

Using symmetry, V = 8 × first-octant volume. Integrate in order z, y, x with limits dependent on the ellipsoid equation to obtain V = (4/3)πabc.

## Answer

**Method.** The ellipsoid x²/a² + y²/b² + z²/c² = 1 is symmetric about all three coordinate planes, so V = 8 × (first octant volume).

In the first octant: 0 ≤ x ≤ a, 0 ≤ y ≤ b√(1 − x²/a²), 0 ≤ z ≤ c√(1 − x²/a² − y²/b²).

```text
V = 8 ∫₀^a ∫₀^{b√(1−x²/a²)} ∫₀^{c√(1−x²/a²−y²/b²)} dz dy dx
```

#### z-integration
The z-integral gives c√(1 − x²/a² − y²/b²) = (c/b)√(k² − y²), where k² = b²(1 − x²/a²).

#### y-integration
```text
(c/b) ∫₀^k √(k² − y²) dy = (c/b)(πk²/4)
                         = (c/b)(π/4) b²(1 − x²/a²)
                         = (π b c/4)(1 − x²/a²)
```

#### x-integration
```text
V = 8 (π b c/4) ∫₀^a (1 − x²/a²) dx
  = 2π b c [ x − x³/(3a²) ]₀^a
  = 2π b c (2a/3)
  = 4π abc/3
```

**Answer: V = (4/3)π abc.** For a = b = c this reduces to the sphere (4/3)π a³.

## Check yourself

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

1. Factor used:
   A. 8
   B. 6
   C. 2
   D. 4
   Answer: A. Octants. (Key point: Symmetry)

2. After y integration:
   A. πbc(1−x²/a²)
   B. (πbc/2)(1−x²/a²)
   C. (πbc/4)(1−x²/a²)
   D. (πc/4)(1−x²/a²)
   Answer: C. (c/b)(πk²/4). (Key point: y-integral)

3. Volume:
   A. πabc
   B. (4/3)πabc
   C. (4/3)π(a+b+c)
   D. (2/3)πabc
   Answer: B. Reduces to sphere when a=b=c. (Key point: Result)

## 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

## Related in other subjects

- [Volume of ellipsoid by revolving an ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-ellipsoid-by-ellipse-revolution) — Mathematics I, chapter 2