---
title: "Surface integral over a surface F=3i+xj−yzk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:surface-integral-over-a-surface-f-eight"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/surface-integral-over-a-surface-f-eight
site: BCT Engineering
---

# Surface integral over a surface F=3i+xj−yzk

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

- Asked in: 2068 Chaitra, 5 marks
- Video: [Evaluation of surface integral over the cylinder in first octant](https://www.youtube.com/watch?v=0UguqZ2SEBo) — TEAM Education
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/surface-integral-over-a-surface-f-eight
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∬_S F·n ds where F = 3 i + x j − yz k and S is the surface of the cylinder x² + y² = 9 included in the first octant between the planes z = 0, z = 4.

## Short answer

Parameterize the quarter-cylinder with x=3cosθ, y=3sinθ, dS=3dθdz. F·n = 3cosθ + 3cosθ sinθ. Integrate θ from 0 to π/2 and z from 0 to 4 to obtain 54.

## Answer

S is the quarter of the cylinder x² + y² = 9 with x, y ≥ 0 and 0 ≤ z ≤ 4.

Outward normal: n = (x i + y j)/3. F = 3 i + x j − yz k, so

```text
F·n = (3x + xy)/3
```

Use x = 3cos θ, y = 3sin θ, 0 ≤ θ ≤ π/2, 0 ≤ z ≤ 4, and dS = 3 dθ dz.

```text
∬_S F·n dS = ∫₀⁴ ∫₀^(π/2) (3x + xy) dθ dz
           = ∫₀⁴ ∫₀^(π/2) (9 cos θ + 9 cos θ sin θ) dθ dz
inner: [9 sin θ + (9/2) sin²θ]₀^(π/2) = 9 + 9/2 = 27/2
outer: 4 × 27/2 = 54
```

**Answer: ∬_S F·n dS = 54.**

## Check yourself

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

1. S is?
   A. Quarter of cylinder, x,y>=0, 0<=z<=4
   B. Whole cylinder
   C. Disc
   D. Sphere
   Answer: A. First octant. (Key point: Surface)

2. F.n?
   A. 3x+xy
   B. (3x + xy)/3
   C. (3x-yz)/3
   D. x
   Answer: B. Dot with n. (Key point: F.n)

3. dS?
   A. dθ dz
   B. 9 dθ dz
   C. 3 dθ dz
   D. r dr dθ
   Answer: C. a dθ dz with a=3. (Key point: Area element)

4. Flux?
   A. 27/2
   B. 108
   C. 18
   D. 54
   Answer: D. 4 x 27/2. (Key point: Result)

## Related in Mathematics III

- [Surface integral over a surface F=yzi+zxj+xyk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asurface-integral-over-a-surface-f) — Mathematics III, chapter 2, asked 5×
- [Surface integral over a surface F=xyi−x²j+(x+z)k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asurface-integral-over-a-surface-f-twelve) — Mathematics III, chapter 2
- [Surface integral over a surface F=xi+yj+zk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asurface-integral-over-a-surface-f-ten) — 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×
- [Surface integral over a surface F=6zi−4j+yk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asurface-integral-over-a-surface-f-five) — Mathematics III, chapter 2, asked 2×
- [Surface integral over a surface F=x²i+y²j+z²k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asurface-integral-over-a-surface-f-two) — Mathematics III, chapter 2, asked 3×

## Related in other subjects

- [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×
- [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 ellipse in first quadrant](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-xy-over-ellipse) — Mathematics, chapter 2, asked 3×
- [Triple integral x² over tetrahedron](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Atriple-integral-x2-over-tetrahedron) — Mathematics, chapter 2, asked 2×