Compute the closed curve integral of F (dot) dR in two ways.....

Let C be the square with vertices (-1,0), (0,0), (0,1), and (-1,1). Let F=[ycos(x)]i -[y^3] j. Compute the closed curve integral of F (dot) dR in two ways:

once using a double integral

and once by summing four line integrals (one for the each edge of the square)

Please help! I am really struggling

Re: Compute the closed curve integral of F (dot) dR in two ways.....

Evaluating using line integral:

F.dr= ycos x dx - y^3dy

C the boundary of the square = C1+C2+C3+C4

Along each Ci, one of the variable x or y is a constant. Hence your integration becomes easy.

Use Green's theorem to convert into double integral.

Hope this helps!