Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise The circulation line integral of Fequalsleft angle 4 xy squared comma 2 x cubed plus y right angle where C is the boundary of StartSet (x comma y ): 0 less than or equals y less than or equals sine x comma 0 less than or equals x less than or equals pi EndSet

The answer is

Step-by-step explanation:

Recall that green theorem is as follows: Given a field F(x,y) = (P(x,y),Q(x,y)) and a closed curve C that is counterclockwise oriented. If P and Q are continuosly differentiable, then

where R is the region enclosed by the curve C.

In this particular case, we have the following field . We are given the description of the region R as . So, in this case (calculations are omitted)

Thus,

So,

Since , then

Consider the integrals

Then, by using integration by parts (whose calculations are omitted) we get

Then, we have that

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step-by-step explanation:

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