---
title: "Triple integral x dv over tetrahedron"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:triple-integral-x-dv-over"
source: https://www.bctengineer.com/q/semester_2/mathematics/triple-integral-x-dv-over
site: BCT Engineering
---

# Triple integral x dv over tetrahedron

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2072 Ashwin, 5 marks
- Video: [Tetrahedron Region In Triple Integrals || Multiple Integrals ||](https://www.youtube.com/watch?v=zrA5w00tkAw) — Tutor Abhi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/triple-integral-x-dv-over
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate: ∭_V x dv where V is bounded by the coordinate planes and the plane x+y+z = 1.

## Short answer

The region is the tetrahedron bounded by the coordinate planes and x+y+z=1. Integrate x dz dy dx with limits 0≤z≤1−x−y, 0≤y≤1−x, 0≤x≤1. Result: 1/24.

## Answer

**Region.** The tetrahedron bounded by x = 0, y = 0, z = 0 and x + y + z = 1: 0 ≤ z ≤ 1 − x − y, 0 ≤ y ≤ 1 − x, 0 ≤ x ≤ 1.

```text
I = ∫₀¹ ∫₀^{1−x} ∫₀^{1−x−y} x dz dy dx
```

#### z-integration
∫₀^{1−x−y} x dz = x(1 − x − y).

#### y-integration
```text
∫₀^{1−x} x(1 − x − y) dy = x [ (1−x)y − y²/2 ]₀^{1−x}
                         = x (1−x)²/2
```

#### x-integration
```text
I = (1/2) ∫₀¹ x(1 − x)² dx
  = (1/2) ∫₀¹ (x − 2x² + x³) dx
  = (1/2)(1/2 − 2/3 + 1/4)
  = (1/2)(1/12)
  = 1/24
```

(Check by Beta function: ∫₀¹ x(1−x)² dx = B(2,3) = 1!·2!/4! = 1/12.)

**Answer: 1/24.**

## Check yourself

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

1. y runs:
   A. 0 to 1
   B. 0 to x
   C. 0 to 1−x−y
   D. 0 to 1−x
   Answer: D. Projection. (Key point: Limits)

2. Integrand:
   A. (1−x)²/2
   B. x²(1−x)
   C. x(1−x)²/2
   D. x(1−x)
   Answer: C. x[(1−x)y − y²/2]. (Key point: After y)

3. Value:
   A. 1/6
   B. 1/24
   C. 1/12
   D. 1/8
   Answer: B. (1/2)B(2,3)=1/24. (Key point: Result)

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