---
title: "Evaluate using Gauss' divergence theorem F=(2xy+z)i+y²j−(x+3y)k"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-gauss-divergence-theorem-f-nine"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-nine
site: BCT Engineering
---

# Evaluate using Gauss' divergence theorem F=(2xy+z)i+y²j−(x+3y)k

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2076 Asoj, 5 marks
- Video: [Gauss divergence theorem](https://www.youtube.com/watch?v=PfBdFs51trI) — Kamalraj Natesan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-nine
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Use Gauss' divergence theorem to evaluate ∬_S F·n ds where F = (2xy + z) i + y² j − (x + 3y) k and S is the surface bounded by the plane 2x + 3y + z = 6, x = 0, y = 0, z = 0.

## Short answer

div F = 4y. By the divergence theorem, the flux is the triple integral of 4y over the tetrahedron bounded by 2x+3y+z=6 and coordinate planes, which evaluates to 12.

## Answer

**Gauss' divergence theorem:** ∬_S F·n dS = ∭_V div F dV. S is the closed surface of the tetrahedron bounded by the plane 2x + 3y + z = 6 and the coordinate planes (vertices (3,0,0), (0,2,0), (0,0,6)).

```text
F = (2xy + z) i + y² j − (x + 3y) k
div F = 2y + 2y + 0 = 4y
```

Limits: x from 0 to 3; y from 0 to (6 − 2x)/3; z from 0 to 6 − 2x − 3y.

```text
∭ 4y dV = ∫₀³ ∫₀^((6−2x)/3) 4y (6 − 2x − 3y) dy dx
```

Let k = 6 − 2x. The inner integral is

```text
∫₀^(k/3) (4ky − 12y²) dy = [2ky² − 4y³]₀^(k/3)
                         = 2k·k²/9 − 4k³/27
                         = 2k³/9 − 4k³/27 = 2k³/27
```

Now integrate with respect to x:

```text
∫₀³ (2k³/27) dx = (2/27) ∫₀³ (6 − 2x)³ dx
                = (2/27) · [−(6 − 2x)⁴/8]₀³
                = (2/27) · (1296/8)
                = (2/27)(162) = 12
```

**Answer: ∬_S F·n dS = 12.**

## Check yourself

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

1. div F?
   A. 2y
   B. 4
   C. 4y
   D. y
   Answer: C. 2y + 2y. (Key point: Divergence)

2. Plane 2x+3y+z=6 intercepts?
   A. (6,0,0),(0,4,0),(0,0,2)
   B. (3,0,0),(0,3,0),(0,0,6)
   C. (2,0,0),(0,3,0),(0,0,6)
   D. (3,0,0),(0,2,0),(0,0,6)
   Answer: D. Set pairs to zero. (Key point: Vertices)

3. y goes from 0 to?
   A. (6-2x)/3
   B. 3-x
   C. 6-2x
   D. 2
   Answer: A. From z = 0 on the plane. (Key point: y-limit)

4. Flux?
   A. 27
   B. 12
   C. 6
   D. 24
   Answer: B. Computed integral. (Key point: Result)

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