use green's theorem to evaluate the integral of x^2dy where C is the boundary of the rectangle with vertices (0,0),(2,0),(2,3),(0,3) oriented counterclockwise.
the answer is 12. can someone please explain
To use Green's theorem to evaluate the given line integral, we'll need to convert it into a double integral over the region bounded by the curve. Green's theorem states that for a vector field F = (M,N), oriented counterclockwise, the line integral around a closed curve C is equal to the double integral of the curl of F over the region R bounded by C.
In this case, we are given the line integral of x^2dy over the rectangle with vertices (0,0), (2,0), (2,3), and (0,3), oriented counterclockwise.
To apply Green's theorem, we need to find the vector field F = (M, N) such that the line integral of F · dr (dot product) gives us the integrand x^2dy.
Since F = (M, N), the line integral becomes
∫ F · dr = ∫ (M dx + N dy)
Now, let's consider the two components of F and find their partial derivatives:
M = 0 (no x-component in F)
N = x^2 (y-component)
To find the curl of F, we calculate the cross partial derivatives:
∂N/∂x = 0
∂M/∂y = 0
Since the curl of F is zero, Green's theorem tells us that the line integral around the boundary of the region is zero:
∫ F · dr = ∬ curl(F) dA = 0
Therefore, the given line integral ∫ x^2 dy is equal to zero.
Hence, the answer should be zero, not 12.