---
title: "Evaluate a line integral F=sinyi+x(1+cosy)j"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-a-line-integral-f-sinyi"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-f-sinyi
site: BCT Engineering
---

# Evaluate a line integral F=sinyi+x(1+cosy)j

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

- Asked in: 2082 Shrawan, 5 marks; 2081 Baishakh, 5 marks; 2080 Baishakh, 5 marks; 2078 Bhadra, 5 marks; 2075 Chaitra, 5 marks; 2074 Chaitra, 5 marks (asked 6 times)
- Video: [Evaluate the integral ∫F.dr where F=(siny)i+x(1+cosy)j and the curve C is the circular path given by](https://www.youtube.com/watch?v=aCTbs0Zmowg) — Doctor of Mathematics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-f-sinyi
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Given the vector field F = sin y i + x(1 + cos y) j, evaluate the line integral over the circular path given by x² + y² = a², z = 0.

## Short answer

By Green’s theorem, ∮_C F·dr = ∬_R (∂Q/∂x − ∂P/∂y) dx dy. Here ∂Q/∂x − ∂P/∂y = 1, so the integral equals the area of the disc, πa².

## Answer

Here F = sin y i + x(1 + cos y) j, so with F = P i + Q j:

```text
P = sin y ,    Q = x(1 + cos y)
```

The path is the circle C: x² + y² = a², z = 0, which is a simple closed curve (taken counter-clockwise). The integral is ∮_C F·dr = ∮ (P dx + Q dy).

#### Use Green's theorem
```text
∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dx dy
```

Compute the derivatives:
- ∂Q/∂x = 1 + cos y
- ∂P/∂y = cos y

```text
∂Q/∂x − ∂P/∂y = (1 + cos y) − cos y = 1
```

So

```text
∮_C F·dr = ∬_R 1 dx dy = area of the disc x² + y² ≤ a² = πa²
```

**Answer: ∮_C F·dr = πa²** (counter-clockwise; −πa² if the circle is traversed clockwise).

#### Direct check (numerical)
Putting x = a cos t, y = a sin t, 0 ≤ t ≤ 2π, the integral becomes ∫₀^{2π} [ −a sin t · sin(a sin t) + a² cos²t (1 + cos(a sin t)) ] dt. For a = 1 this evaluates to 3.14159 = π, and for a = 1.5 to 7.0686 = π(1.5)², agreeing with πa².

## Check yourself

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

1. Why use Green's theorem here?
   A. The path is open
   B. The path is a simple closed curve in the plane
   C. F is constant
   D. The curve is a line
   Answer: B. A closed circle in z=0 suits Green's theorem. (Key point: Choice of method)

2. ∂Q/∂x and ∂P/∂y with P=sin y, Q=x(1+cos y):
   A. 1+cos y and cos y
   B. cos y and 1+cos y
   C. 1 and 0
   D. x and cos y
   Answer: A. ∂Q/∂x = 1+cos y; ∂P/∂y = cos y. (Key point: Derivatives)

3. ∂Q/∂x − ∂P/∂y equals:
   A. cos y
   B. 0
   C. 1+2cos y
   D. 1
   Answer: D. Difference is 1. (Key point: Integrand)

4. The integral (counter-clockwise) is:
   A. 2πa
   B. 0
   C. πa²
   D. πa
   Answer: C. The double integral of 1 is the disc area. (Key point: Answer)

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