Sketch the vector field f(x, y)=−5i+j and calculate the line integral of f along the line segment from (3,5) to (−3,1).

Parameterize the line segment by



with . Then the line integral is given by

.

We can also use the fact that



so that the gradient theorem applies, which means the integral is path independent and the value of the line integral can be found by evaluating

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