Question

Let \( R \) be a region in the first quadrant of the \( xy \)-plane enclosed by a closed curve \( C \) considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region \( R \)?

• A. ( \int \int_R dx \, dy \)
• B. ( \oint_C x \, dy \)
• C. ( \oint_C y \, dx \)
• D. ( \frac{1}{2} \oint_C (x \, dy - y \, dx) \)

Hint:

Green's Theorem is often used to convert a line integral around a closed curve into a double integral for area or other quantities. The formula \( \frac{1}{2} \oint_C (x \, dy - y \, dx) \) is a standard expression for area.

Answer 1

The Correct Option is A

Step 1: Understanding the expressions for area.
The area of a region \( R \) enclosed by a closed curve \( C \) in the plane can be computed using Green's Theorem. Green's Theorem states that: \[ \text{Area} = \frac{1}{2} \oint_C (x \, dy - y \, dx). \] This expression correctly represents the area of the enclosed region \( R \).
Step 2: Analyzing the options.
- (A) \( \int \int_R dx \, dy \): This expression represents the area, but it is not the correct form for a line integral; it’s a double integral for the area. However, it is not the typical line integral representation, so this option does not align with the given options for area calculation.
- (B) \( \oint_C x \, dy \): This expression is not the standard formula for area, as it only involves one variable and is not consistent with Green's Theorem for area calculation.
- (C) \( \oint_C y \, dx \): This is a correct form of an integral used for computing the area when integrated properly in the context of Green's Theorem.
- (D) \( \frac{1}{2} \oint_C (x \, dy - y \, dx) \): This is the correct formula for computing the area using Green's Theorem.
Step 3: Conclusion.
The correct answers are (A) and (B), as these do not represent the standard expressions for calculating the area using Green's Theorem.