Question

Evaluate the following integrals using Green's theorem

(a) ∮[C] 2x sin(y) dx + x^2 cos(y) dy
where C is the polygon with vertices (0,0), (2,0), (3,1), (1,3)

Answer 1

To evaluate the given line integral using Green's theorem, we first find the partial derivatives of the given functions. Then, we apply Green's theorem to convert the line integral into a double integral over the region bounded by the curve. Finally, we evaluate the double integral to find the value of the line integral.

Step-by-step explanation:

To evaluate the integral using Green's theorem, we need to find the partial derivatives of the given functions. Let's denote the first function as F = 2x sin(y) and the second function as G = x^2 cos(y). We calculate the partial derivatives of F and G with respect to x and y.

Now, we can apply Green's theorem to evaluate the line integral. Green's theorem states that the line integral along a closed curve C can be evaluated as a double integral over the region bounded by C. The line integral is equal to the double integral of the curl of the vector field (G, -F) with respect to the variables x and y.

Let's calculate the curl of the vector field (G, -F):

Curl(G, -F) = (∂G/∂y - ∂(-F)/∂x, ∂(-F)/∂y - ∂G/∂x) = (∂G/∂y + ∂F/∂x, ∂F/∂y - ∂G/∂x)

Now, we integrate this curl over the given polygon C to evaluate the line integral.