Evaluate the integral ∮cf⋅dr for f=(3x2y)i+(2x2−3xy2)j on the curve c consisting of the x-axis from x=0 to x=2, the arc of the circle x2+y2=4 up to the line y=x, and the line y=xdown to the origin.

Write the line integral as



In other words, , and .

The curve is closed, so we can apply Green's theorem:





Convert to polar coordinates to write this as

In all honesty, there really is no “work” per say to show. mainly it’s just using the process of elimination. the question is asking which domain(s) would be the most appropriate for the function c(n). n is the number of observed vehicles so right away, 1, 2, and 3 can be eliminated.why? 1, 2 start in the negatives and therefore it is impossible to observe negative cars. so, #4 would be the best answer to the given problem.