To find ∫ (x − y) dx + (x + y) dy directly, we must parameterize C. Since C is a circle with radius 2 centered at the origin, then a parameterization is the following. (Use t as the independent variable.) x = 2 cos(t)
y = 2 sin(t)
0 ≤ t ≤ 2π

With this parameterization, find the followings

dy=
dx=