Final answer:
Green's Theorem is applied by sketching the triangular path and converting the line integral into a double integral over the triangular region. We compute the integral from 0 to 4 for x and 0 to x for y, using the expression 2x - 2y from the partial derivatives of P and Q.
Step-by-step explanation:
To use Green's Theorem to evaluate the work done along a path defined by a triangle with vertices (0,0), (4,0), and (0,4), with the vector field given by P(x, y) = y² and Q(x, y) = x², first sketch the triangular path, which is a right-angled triangle in the first quadrant.
Green's Theorem can be used to convert the line integral around a closed curve C to a double integral over the region D that C encloses. Green's Theorem states:
∫C P dx + Q dy = ∫D ( Qx - Py ) dA
Where P = y², Q = x², Qx = 2x, and Py = 2y. Computing Qx - Py gives 2x - 2y. The work done by the vector field is then the area integral of 2x - 2y over the triangular region.
Setting up the integral, we have:
Since the triangle is right-angled, we can integrate y from 0 to x (since y = 4 - x along the hypotenuse) and x from 0 to 4. The computed integral gives us the work done.