Area with Parametric Equations — Question 5

Question 5

Problem

A simple, piecewise smooth closed curve is traversed once clockwise. Explain why $\int y\,dx$ may be positive even though area cannot be negative.

Question 5 – Solution

Solution

1. A line integral around a closed curve is signed: reversing the direction of traversal reverses the sign because $dx$ changes sign.

2. For a positively oriented, counterclockwise curve $C$, Green’s theorem gives $$A=-\oint_C y\,dx.$$ Thus $\oint_C y\,dx=-A$ for counterclockwise traversal.

3. If the same boundary is traversed clockwise, the line integral changes sign: $$\oint_{-C}y\,dx=-\oint_Cy\,dx=A.$$ Therefore, $\int y\,dx$ can be positive for clockwise motion; the sign describes orientation, not a negative or positive physical area by itself.

4. A direction-independent formula for geometric area is $$\boxed{A=\left|\oint_C y\,dx\right|}.$$ Equivalently, use $-\oint_Cy\,dx$ with counterclockwise orientation or reverse the bounds when necessary.