Calculate the integral along the curve C of capital F dotted with DR, where capital F of x comma y equals open angle bracket sine x times sine y comma 5 minus cosine x times cosine y close angle bracket. And C is a semicircle with starting 0 comma pi and end point 0 comma negative pi. First, let's find a potential function by taking the antiderivative, with respect to x and y, for each of these components.

And a potential function is lowercase f of x comma y equal to negative cosine x times sine y plus 5y minus-- well, minus cosine x sine y. Actually, that means we can just leave that as just negative cosine x times sine y plus 5y. I'm just going to go ahead and take the derivative of this with respect to x.

That would be sine x, sine y. So that would be the same x component. And the y component, that would be 5 minus cosine x cosine y. Yeah, that's actually a good potential function.

Let's go ahead and find this integral. We already know the endpoint. So this integral, along the curve C of capital F dotted with a dr, by the fundamental theorem of line integrals, this would be equal to the integral.

It'd be equal to lowercase f, evaluated at 0 comma negative pi minus f evaluated at 0 comma pi. Both of those will actually be negative 5 pi. This is equal to negative 5 pi minus 5 pi.

Sorry, the first will be negative 5 pi. The second evaluation will be 5 pi. So subtracting those two will be negative 10 pie.