INSTRUCTOR: Given that f of x comma y, equal to parentheses x minus 1, close parentheses, squared, times y, plus, open parentheses, y plus 1, closed parentheses, squared, times x, it's a potential function for F of x comma y, equal to, angle bracket, 2xy minus 2y, plus open parentheses y plus 1, close parentheses, squared, comma, open parentheses, x minus 1, close parentheses, squared, plus 2yx, plus 2x, close angle bracket. Calculate the integral along the curve, C, of F, dotted with dr, where C is the lower 1/2 of the unit circle oriented counterclockwise.

First, our parameterization for the curve, C, being the unit circle is r of t equal to, angle bracket, cosine t, comma sine t. And in order to make this the lower 1/2 of the unit circle, we'll restrict this to pi less than or equal to t, less than or equal to 2 pi. Now what we actually want to find is the integral from pi to 2 pi with respect to r. Or, really, with respect to time, of F dotted with dr.

By the fundamental theorem of line integrals-- yes, all right-- that is going to be equal to the integral from pi to 2 pi of the gradient of f dotted with dr, which will be equal to f of r of 2 pi, close parentheses, minus f, open parentheses, r of pi, close parentheses, which is the equivalent of saying f of the point 1, 0. And this is the f I'm talking about here. Lowercase f evaluated at 1, 0, minus f evaluated at the point -1, 0, is the two endpoints points of C, that curve, and that equals 2.