Suppose C is a positively oriented, piecewise-smooth, simple closed curve. Verify that the result of Green's Theorem holds for the line integral ry° dz - a'y dy, r’y dy, where C consists of the parabola y = x² from (-1,1) to (1,1) and the line segment from (1,1) to (-1,1).

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A: We will find out the required solution.

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A: by usingGreen's Theorem to evaluate the line integral along the given positively oriented curve.…

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A: We will evaluate the given line integral.

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A: We will evaluate the given integral.

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A: According to the problem, we have

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A: Given line integral is,

Q: |[-y dx + xy dy]

A: See the attachment

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A: See attached file for a step by step explanation to evaluate the given line integral.

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A: We know that r=xi+yj As, the path followed is y=x2 So, r=xi+x2j Let x=t, then  r=ti+t2jdrdt=i+2tj…

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