Use Green's Theorem to evaluate ?C F·dr. (Check the orientation of the curve before applying the theorem.) F(x, y) =< x + 4y3, 4x2 + y>

C consists of the arc of the curve y = sin x from (0, 0) to (p, 0) and the line segment from (p, 0) to (0, 0).

Step-by-step explanation:

given a field of the form F = (P(x,y),Q(x,y) and a simple closed curve positively oriented, then

where A is the area of the region enclosed by C.

In this case, by the description we can assume that C starts at (0,0). Then it goes the point (pi,0) on the path giben by y = sin(x) and then return to (0,0) along the straigth line that connects both points. Note that in this way, the interior the region enclosed by C is always on the right side of the point. This means that the curve is negatively oriented. Consider the path C' given by going from (0,0) to (pi,0) in a straight line and the going from (pi,0) to (0,0) over the curve y = sin(x). This path is positively oriented and we have that

We use the green theorem applied to the path C'. Taking we get

A is the region enclosed by the curves y =sin(x) and the x axis between the points (0,0) and (pi,0). So, we can describe this region as follows

This gives use the integral

Integrating accordingly, we get that

So

b. x + 1

step-by-step explanation:

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