Answer:
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