---
title: "Volume of ellipsoid by revolving an ellipse"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:volume-ellipsoid-by-ellipse-revolution"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/volume-ellipsoid-by-ellipse-revolution
site: BCT Engineering
---

# Volume of ellipsoid by revolving an ellipse

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2069 Asar
- Video: [Volume of the Solid Generated by Rotating An Ellipse Around The x-axis](https://www.youtube.com/watch?v=gSZWweEQvno) — The Math Sorcerer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/volume-ellipsoid-by-ellipse-revolution
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Find the volume of the ellipsoid formed by the revolution of an ellipse x²/a² + y²/b² = 1.

## Short answer

Revolve about x-axis: V = 4πab²/3 cubic units. Revolve about y-axis: V = 4πa²b/3 cubic units, using V = π∫ y² dx or π∫ x² dy respectively.

## Answer

Ellipse: x²/a² + y²/b² = 1. Revolving it about an axis gives an ellipsoid of revolution (a spheroid).

#### About the x-axis (prolate spheroid when a > b)
The solid is symmetrical about the y-axis, so V = π ∫_(−a)^a y² dx = 2π ∫₀^a y² dx, with y² = b²(1 − x²/a²):

```text
V = 2π b² ∫₀^a (1 − x²/a²) dx
  = 2π b² [x − x³/(3a²)]₀^a
  = 2π b² (a − a/3)
```
**V = 4πab²/3.**

#### About the y-axis (oblate spheroid when a > b)
Similarly V = 2π a² ∫₀^b (1 − y²/b²) dy = 2πa² (b − b/3), so

**V = 4πa²b/3.**

For a sphere (a = b = r) both give 4πr³/3, as expected.

## Check yourself

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

1. Revolving about the x-axis gives
   A. 4πa^2b/3
   B. 4πab^2/3
   C. 2πab^2/3
   D. 4πab/3
   Answer: B. 2πb^2(a - a/3). (Key point: About x-axis)

2. Revolving about the y-axis gives
   A. 4πab^2/3
   B. 2πa^2b/3
   C. 4πa^2 b/3
   D. 4πab/3
   Answer: C. 2πa^2(b - b/3). (Key point: About y-axis)

3. For a = b = r both give
   A. 2πr^3/3
   B. πr^3
   C. 4πr^2
   D. 4πr^3/3
   Answer: D. Sphere volume. (Key point: Sphere check)

## Related in Mathematics I

- [Volume by revolving ellipse about x=2a](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-ellipse-about-x-2a) — Mathematics I, chapter 2
- [Volume of cardioid r=a(1+cosθ) about initial line](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-cardioid-1-plus-cos-initial-line) — Mathematics I, chapter 2, asked 6×

## Related in other subjects

- [Volume of ellipsoid by triple integration](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Avolume-ellipsoid-triple-integration) — Mathematics, chapter 2