---
title: "Volume by revolving ellipse about x=2a"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:volume-ellipse-about-x-2a"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/volume-ellipse-about-x-2a
site: BCT Engineering
---

# Volume by revolving ellipse about x=2a

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2070 Asar
- Video: [The volume of solid obtained by revolving the area of the ellipse x2a2+y2b2=1|m1|btech|degree|](https://www.youtube.com/watch?v=IrFpqsGSe0I) — RS ACADEMY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/volume-ellipse-about-x-2a
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Show that the volume of the solid formed by revolving the ellipse x²/a² + y²/b² = 1 about the line x = 2a is 4π²a²b cubic units.

## Short answer

Using washers perpendicular to the axis x=2a with outer radius 2a+x₁ and inner radius 2a-x₁ gives V = π∫(R²-r²)dy = 8πa²∫√(1-y²/b²)dy = 4π²a²b cubic units.

## Answer

Ellipse: x²/a² + y²/b² = 1. The line x = 2a is parallel to the y-axis and does not meet the ellipse (the ellipse reaches only x = a), so the solid is a ring (torus-like) shape. Use the washer method, slicing perpendicular to the axis of rotation (horizontal slices at height y).

At height y, the horizontal strip runs from x = −x₁ to x = +x₁ where x₁ = a√(1 − y²/b²). Distances from the line x = 2a:

- outer radius R = 2a + x₁
- inner radius r = 2a − x₁

```text
V = π ∫_(−b)^b (R² − r²) dy
  = π ∫_(−b)^b 8a x₁ dy
  = 8πa² ∫_(−b)^b √(1 − y²/b²) dy
```
Since ∫_(−b)^b √(b² − y²) dy = πb²/2 (the area of a semicircle of radius b), ∫_(−b)^b √(1 − y²/b²) dy = (1/b)(πb²/2) = πb/2.

V = 8πa² · πb/2

**V = 4π²a²b cubic units.** Shown.

(Check by Pappus's theorem: area of ellipse πab times the path of its centre 2π(2a) gives 4π²a²b.)

## Check yourself

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

1. The ellipse revolved about x = 2a gives a ring because
   A. the line passes through the centre
   B. the line does not touch the ellipse
   C. the ellipse is a circle
   D. the axis is horizontal
   Answer: B. The ellipse reaches only x = a. (Key point: Shape)

2. The outer and inner radii are
   A. 2a and x1
   B. a + x1 and a - x1
   C. 2a + x1 and 2a - x1
   D. x1 and 2a
   Answer: C. Distances from the line. (Key point: Radii)

3. R^2 - r^2 equals
   A. 4a x1
   B. 4a^2
   C. 2a x1
   D. 8a x1
   Answer: D. (2a+x1)^2 - (2a-x1)^2 = 8a x1. (Key point: Difference)

4. The volume is
   A. 4π^2 a^2 b
   B. 2π^2 a^2 b
   C. 4π a^2 b
   D. 8π^2 a b^2
   Answer: A. Pappus: πab · 2π(2a). (Key point: Result)

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