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Calculus IIIChapter 5: Line IntegralsSection 5-3

Line Integrals - Part II — Question 5

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Question 5

Let CC be the boundary of the rectangle 0≤x≤3,0≤y≤2,0\le x\le 3,\qquad 0\le y\le 2, traversed counterclockwise. Evaluate ∮Cydx\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.

Original worksheet page 1: question and worked solution for 5-3-005
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Question 5 – Solution

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

Step 1: Traverse the edges

See the diagram in the original worksheet below.

Along the bottom edge, y=0y=0, so its contribution is 00. On the right and left edges, xx is constant, so dx=0dx=0 and both contributions are also 00.

Step 2: Top edge Counterclockwise travel moves along the top from (3,2)(3,2) to (0,2)(0,2). Thus ∫topydx=∫302dx=2[x]30=−6.\int_{\text{top}}y\,dx =\int_3^0 2\,dx =2[x]_3^0=-6.

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

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

Original worksheet page 2: question and worked solution for 5-3-005

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

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