Verify Greens theorem in the plane for Q ((xy+y²)dx+x²dy) where C is the curve of region y=x² and y=x

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Greens theorem states that for any simple, closed curve C in the plane, and for any region D bounded by C, the following relationship holds:

∫C Q dx + ∫C Q dy = ∫∫D (∂Q/∂y – ∂Q/∂x) dxdy

To verify Greens theorem for the given function Q, we can apply the theorem to the curve C and region D defined by the equation y=x² and y=x, respectively.

For the given function Q, we have:

Q = (xy+y²)dx+x²dy

Therefore, the partial derivatives of Q with respect to x and y are:

∂Q/∂x = ydx + 0dy = ydx

∂Q/∂y = xdx + 2ydy = xdx + 2ydy

Substituting these partial derivatives into Greens theorem, we get:

∫C Q dx + ∫C Q dy = ∫∫D (∂Q/∂y – ∂Q/∂x) dxdy

= ∫C (xy+y²)dx + ∫C x²dy = ∫∫D (xdx + 2ydy – ydx) dxdy

We can now evaluate the integrals on the right-hand side of the equation to verify that Greens theorem holds for the given function Q and curve C.