---
title: "Triple integral x² over tetrahedron"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:triple-integral-x2-over-tetrahedron"
source: https://www.bctengineer.com/q/semester_2/mathematics/triple-integral-x2-over-tetrahedron
site: BCT Engineering
---

# Triple integral x² over tetrahedron

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2074 Bhadra, 5 marks; 2069 Bhadra, 5 marks (asked 2 times)
- 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-x2-over-tetrahedron
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate ∭ x² dx dy dz over the region v bounded by the planes x = 0, y = 0, z = 0 and x+y+z = a.

## Short answer

Tetrahedron bounded by x=0,y=0,z=0, x+y+z=a. Integrate in order z, y, x: I = ∫₀^a ∫₀^{a−x} ∫₀^{a−x−y} x² dz dy dx, which evaluates to a⁵/60.

## Answer

**Region.** The tetrahedron bounded by the coordinate planes and x + y + z = a: x ≥ 0, y ≥ 0, 0 ≤ z ≤ a − x − y, 0 ≤ y ≤ a − x, 0 ≤ x ≤ a.

```text
I = ∫₀^a ∫₀^{a−x} ∫₀^{a−x−y} x² dz dy dx
```

#### Integrate in z
∫₀^{a−x−y} x² dz = x²(a − x − y).

#### Integrate in y
```text
∫₀^{a−x} x²(a − x − y) dy = x² [ (a−x)y − y²/2 ]₀^{a−x}
                          = x² (a − x)²/2
```

#### Integrate in x
```text
I = (1/2) ∫₀^a x²(a − x)² dx
  = (1/2) ∫₀^a (a²x² − 2a x³ + x⁴) dx
  = (1/2) ( a²·a³/3 − 2a·a⁴/4 + a⁵/5 )
  = (a⁵/2)(1/3 − 1/2 + 1/5)
  = (a⁵/2)(1/30)
  = a⁵/60
```

**Answer: a⁵/60.**

## Check yourself

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

1. The limit for z is:
   A. 0 to a−x
   B. 0 to a+x+y
   C. 0 to a−x−y
   D. 0 to a
   Answer: C. Plane x+y+z = a. (Key point: Limits)

2. After z and y integration the integrand is:
   A. x²(a−x)²
   B. x(a−x)²/2
   C. x²(a−x)
   D. x²(a−x)²/2
   Answer: D. x²[(a−x)y − y²/2]. (Key point: After y)

3. The value is:
   A. a⁵/30
   B. a⁵/20
   C. a⁴/60
   D. a⁵/60
   Answer: D. (a⁵/2)(1/30). (Key point: Result)

## Related in Mathematics

- [Triple integral (2x+y) over cylinder region](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Atriple-integral-2x-y-over) — Mathematics, chapter 2
- [Double integral xy over region bounded by x²=4ay](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-xy-over-region) — Mathematics, chapter 2
- [Triple integral x dv over tetrahedron](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Atriple-integral-x-dv-over) — Mathematics, chapter 2
- [Double integral y over region between two parabolas](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-y-over-region) — Mathematics, chapter 2
- [Evaluate ∫∫ e^(x²) dx dy](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-x2-dx-dy) — Mathematics, chapter 2
- [Dirichlet integral over positive octant region](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adirichlet-integral-over-positive-octant) — Mathematics, chapter 2, asked 3×

## Related in other subjects

- [Define complex integral; evaluate ∮log z dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-log-z) — Applied Mathematics, chapter 1
- [Evaluate ∮(z+1)dz over a square](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-square-contour) — Applied Mathematics, chapter 1
- [Volume integral over a region F=2xzi−xj+y²k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Avolume-integral-over-a-region-f) — Mathematics III, chapter 2
- [Surface integral over a surface F=4xi−2y²j+z²k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asurface-integral-over-a-surface-f-three) — Mathematics III, chapter 2, asked 2×