Question

To evaluate the line integral f 2 sin y d + ² cos y dy using Green's theorem, where C is the polygon with vertices (0,0), (2,0), (3,1), and (1,3), we need to calculate the double integral over the enclosed region. What is the result of this double integral?

Answer 1

To evaluate the given line integral of 2 sin y + ² cos y , Green's Theorem is applied, revealing that the double integral of the curl over the enclosed region is zero, indicating the line integral value is also zero.

Step-by-step explanation:

To evaluate the line integral 2 sin y dx + 2 cos y dy using Green's Theorem, we convert the line integral around the polygon with vertices (0,0), (2,0), (3,1), and (1,3) into a double integral over the area enclosed by the polygon. Green's Theorem relates the circulation of a vector field around a closed path C to the double integral of the curl of the vector field over the region D enclosed by C. Since the line integral given is already in the form P dx + Q dy, where P = 2 sin y and Q = 2 cos y, we can apply Green's Theorem directly.

The curl of the vector field ∇ × F is ∂Q/∂x - ∂P/∂y. Here, P and Q do not depend on x, so the partial derivatives ∂Q/∂x and ∂P/∂y are both zero, and the curl is zero. Therefore, the double integral of the curl over the area D is also zero, which means the original line integral is zero.