Suppose f⃗ (x, y)=xyi⃗ +(x−y)j⃗ and c is the triangle from (4,0) to (−4,0) to (0,4) to (4,0). (a) find the line integral of f⃗ along each segment of the triangle

The line integral of f along each segment of the triangle is given below and this can be determined by integrating each segment of the triangle.

Given :

First, parametrize each segment of the triangle.

where

Now, the line integral of f along each segment of the triangle is given by:

The line integral of f along each segment of the triangle is given above and this can be determined by integrating each segment of the triangle.

For more information, refer to the link given below:

link

Parameterize each leg by

each with .

The line integrals along each leg (in the same order as above) are

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The total line integral then has a value of -16. We can confirm this by checking with Green's theorem. Notice that as given as clockwise orientation, while Green's theorem assumes counterclockwise. So we must multiply by -1:

as required.

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step-by-step explanation: