Question

Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (−3, 2), (−3, −3), (2, −2), (2, 6), and back to (−3, 2), in that order. Use Green's theorem to evaluate the following integral. (2xy) dx + (xy²) dy

Answer 1

The student's question on evaluating a line integral using Green's theorem is addressed by outlining the application of the theorem to find a double integral over the closed curve C.

Step-by-step explanation:

The student is asked to evaluate the line integral of (2xy) dx + (xy²) dy using Green's theorem over a closed curve C. Green's theorem relates a line integral around a closed curve to a double integral over the region the curve encloses. For the given curve C, we apply Green's theorem by finding the partial derivatives of the functions multiplying dx and dy, that is ∂(2xy)/∂y and ∂(xy²)/∂x, and then integrating the difference of these derivatives over the area enclosed by C. This area is a simple polygon, making the double integral relatively easy to evaluate.