Verify Green's theorem

Green's theorem: ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dx dy.

Here P = 3x² − 8y², Q = 4y − 6xy. C is the closed curve bounded by y = x² and x = y², which meet at (0, 0) and (1, 1). Traverse C counter-clockwise: along y = x² from O(0, 0) to A(1, 1), then back along x = y² from A to O.

Right side: double integral

∂Q/∂x = −6y ,   ∂P/∂y = −16y   ⇒   ∂Q/∂x − ∂P/∂y = 10y

For 0 ≤ x ≤ 1, y runs from x² to √x:

∬ 10y dy dx = ∫₀¹ 5[y²]_{x²}^{√x} dx = 5 ∫₀¹ (x − x⁴) dx = 5(1/2 − 1/5) = 3/2

Left side: line integral

Along y = x² (x: 0 → 1), dy = 2x dx:

∫ [(3x² − 8x⁴) + (4x² − 6x³)(2x)] dx = ∫₀¹ (3x² + 8x³ − 20x⁴) dx = 1 + 2 − 4 = −1

Along x = y² (y: 1 → 0), dx = 2y dy:

∫ [(3y⁴ − 8y²)(2y) + (4y − 6y³)] dy = ∫₁⁰ (6y⁵ − 22y³ + 4y) dy
 = −[ y⁶ − 11y⁴/2 + 2y² ]₀¹ = −(1 − 5.5 + 2) = 5/2

∮_C = −1 + 5/2 = 3/2

Both sides equal 3/2, so Green's theorem is verified. ∎