Apply Green's theorem

Short answer

∂Q∂x − ∂P∂y = −3(x²+y²). In polar coordinates: ∮ = −3 ∫₀²π dθ ∫₀² r³ dr = −3·2π·4 = −24π.

Using Green's theorem, evaluate ∮_C (y³ dx − x³ dy) where C is the boundary of the circle x² + y² = 4.

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

P = y³,  Q = −x³
∂Q/∂x = −3x²,  ∂P/∂y = 3y²
∂Q/∂x − ∂P/∂y = −3(x² + y²)

R is the disc x² + y² ≤ 4. Use polar coordinates: x² + y² = r², dx dy = r dr dθ, 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π.

∮_C = −3 ∫₀^2π ∫₀² r² · r dr dθ
    = −3 · 2π · [r⁴/4]₀²
    = −3 · 2π · 4
    = −24π

Answer: ∮_C (y³ dx − x³ dy) = −24π (C anticlockwise).

✓ Answer checked by Claude Opus 5.5 (high effort)

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