---
title: "Volume common to two cylinders by triple integral"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:volume-common-two-cylinders-triple"
source: https://www.bctengineer.com/q/semester_2/mathematics/volume-common-two-cylinders-triple
site: BCT Engineering
---

# Volume common to two cylinders by triple integral

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2067 Chaitra, 5 marks
- Video: [TRIPLE INTEGRAL Find the volume common to 2 cylinders x^2+y^2 = a^2 and X^2+z^2=a^2](https://www.youtube.com/watch?v=df7mkXlcphc) — m-easy maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/volume-common-two-cylinders-triple
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Find by triple integral, the volume common to the cylinders x²+y² = a² and x²+z² = a².

## Short answer

By symmetry V=8×first-octant volume. Limits: z ≤ √(a²−x²), y ≤ √(a²−x²), x∈[0,a]. ∫₀^a (a²−x²) dx = 2a³/3, so V = 16a³/3 cubic units.

## Answer

**Solid.** The cylinders x² + y² = a² (axis along z) and x² + z² = a² (axis along y) intersect in a solid symmetric about all three coordinate planes. So V = 8 × (first-octant part).

**Limits (first octant).** The solid lies inside both cylinders:
- z runs from 0 to √(a² − x²) (from the second cylinder),
- y runs from 0 to √(a² − x²) (from the first cylinder),
- x runs from 0 to a.

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

#### Integrate
```text
∫₀^{√(a²−x²)} dz = √(a² − x²)

∫₀^{√(a²−x²)} √(a² − x²) dy = (a² − x²)

V = 8 ∫₀^a (a² − x²) dx
  = 8 [ a²x − x³/3 ]₀^a
  = 8 (a³ − a³/3)
  = 16a³/3
```

**Answer: V = 16a³/3.**

## Check yourself

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

1. Why is the volume 8 times the first-octant part?
   A. There are 8 cylinders
   B. The common solid is symmetric about all three coordinate planes
   C. The cylinders meet in 8 curves
   D. Each cylinder has volume 8a^3
   Answer: B. Symmetry about the three planes gives 8 equal parts. (Key point: Use of symmetry)

2. What are the limits in the first octant?
   A. z from 0 to sqrt(a^2 - x^2), y from 0 to sqrt(a^2 - x^2), x from 0 to a
   B. z from 0 to sqrt(a^2 - y^2), y from 0 to a, x from 0 to a
   C. z from 0 to a, y from 0 to a, x from 0 to a
   D. z from 0 to sqrt(a^2 - x^2 - y^2), y from 0 to a, x from 0 to a
   Answer: A. Cylinder x^2+z^2 = a^2 bounds z and cylinder x^2+y^2 = a^2 bounds y, both by sqrt(a^2 - x^2). (Key point: Limits)

3. After integrating in z and then y, what is left to integrate in x?
   A. a^2 - x^2
   B. sqrt(a^2 - x^2)
   C. (a^2 - x^2)^2
   D. a^2 - x
   Answer: A. The z-integral gives sqrt(a^2 - x^2), and the y-integral multiplies by another sqrt(a^2 - x^2). (Key point: After two integrations)

4. What is the volume?
   A. 8a^3/3
   B. 16a^3/9
   C. 16a^3/3
   D. 4 pi a^3/3
   Answer: C. 8 times [a^2 x - x^3/3] from 0 to a = 8(2a^3/3) = 16a^3/3. (Key point: Final volume)

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