Evaluate the line integral, where c is the given curve. ? c sin(xdx cos(ydy c consists of the top half of the circle x2 y2 = 9 from (3, 0 to (-3, 0 and the line segment from (-3, 0 to (-4, 4.

You can parameterize the curve and compute the integral along each component curve, or you can use the fact that is the continuous gradient of a function and observe that the line integral is path independent.

In other words, there is a function such that , so the integral along any curve from the points to is simply



You have , and so



while



where is an arbitrary constant. So we've found that



which means the line integral has a value of