---
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-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-four
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: 2080 Baishakh, 5 marks; 2071 Chaitra, 5 marks (asked 2 times)
- 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-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

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

## Short answer

Divergence ∇·F = 4y. Integrate over the tetrahedron bounded by 2x+2y+z=6 and coordinate planes: the volume integral evaluates to 27.

## Answer

F = (2xy + z) i + y² j − (x + 3y) k. The region V is bounded by the plane 2x + 2y + z = 6 and the coordinate planes x = 0, y = 0, z = 0 (a tetrahedron).

#### Divergence
```text
∇·F = ∂(2xy + z)/∂x + ∂(y²)/∂y + ∂(−x − 3y)/∂z = 2y + 2y + 0 = 4y
```

#### Limits
The plane meets the axes at x = 3, y = 3, z = 6. So

```text
z : 0 → 6 − 2x − 2y ,   y : 0 → 3 − x ,   x : 0 → 3
```

#### Volume integral
```text
∭ 4y dV = ∫₀³ ∫₀^{3−x} 4y (6 − 2x − 2y) dy dx
```

Let m = 3 − x, so 6 − 2x − 2y = 2(m − y):

```text
∫₀^m 8y (m − y) dy = 8 [ m y²/2 − y³/3 ]₀^m = 8 (m³/2 − m³/3) = 4m³/3
```

```text
∫₀³ (4/3)(3 − x)³ dx = (4/3) · [ −(3 − x)⁴/4 ]₀³ = (4/3)(81/4) = 27
```

**Answer: ∬_S F·n dS = 27.**

## Check yourself

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

1. What is div F for F = (2xy+z) i + y^2 j - (x+3y) k?
   A. 4y
   B. 2y
   C. 4y + 1
   D. y
   Answer: A. 2y + 2y + 0 = 4y. (Key point: Divergence)

2. Where does 2x + 2y + z = 6 meet the axes?
   A. x = 6, y = 6, z = 3
   B. x = 3, y = 3, z = 6
   C. x = 2, y = 2, z = 6
   D. x = 3, y = 3, z = 3
   Answer: B. Set two variables to zero in turn. (Key point: Intercepts)

3. What are the limits for the volume?
   A. z: 0 to 6, y: 0 to 3, x: 0 to 3
   B. z: 0 to 3, y: 0 to 6-x, x: 0 to 3
   C. z: 0 to 6-2x-2y, y: 0 to 3-x, x: 0 to 3
   D. x: 0 to 6, y: 0 to 6, z: 0 to 3
   Answer: C. The plane is the upper bound of z over the triangle in the xy-plane. (Key point: Limits)

4. What is the flux?
   A. 54
   B. 9
   C. 81/4
   D. 27
   Answer: D. The integral of (4/3)(3-x)^3 from 0 to 3 equals 27. (Key point: Result)

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