Example 4: Line Integral of a Circle

Find the line integral

where C is the circle x² + y² = 4, shown in Figure 13.2.13.

Figure 13.2.13

We may start at any point of C. Take (2,0) as the initial point. Then C has the parametric equations

x = 2 cos θ, y = 2 sin θ, 0 ≤ θ ≤ 2π.

As θ goes from 0 to 2π, (x, y) goes around C once counterclockwise as required.