Line Integrals - Part II — Question 5

Question 5

Let $C$ be the boundary of the rectangle $$0\le x\le 3,\qquad 0\le y\le 2,$$ traversed counterclockwise. Evaluate $$\oint_C y\,dx$$ by splitting the boundary into four segments.

Tasks

1. Determine which edges contribute zero.

2. Evaluate the contribution from each horizontal edge.

3. Add the pieces and check the sign from the orientation.

Question 5 – Solution

Strategy. On vertical edges $dx=0$, while $y$ is constant on each horizontal edge.

Step 1: Traverse the edges

Along the bottom edge, $y=0$, so its contribution is $0$. On the right and left edges, $x$ is constant, so $dx=0$ and both contributions are also $0$.

Step 2: Top edge Counterclockwise travel moves along the top from $(3,2)$ to $(0,2)$. Thus $$\int_{\text{top}}y\,dx =\int_3^0 2\,dx =2[x]_3^0=-6.$$

Step 3: Sum $$\boxed{\oint_Cy\,dx=0+0-6+0=-6}.$$

Verification The only nonzero edge has $y>0$ and moves left, so $dx<0$; a negative answer is necessary. Reversing the rectangle would change the answer to $+6$.