calculus green theorem

Question

Use Green's Theorem to evaluate ∫CF⋅nds, where F=⟨√x+5y,4x+5y⟩ and C is the boundary of the region enclosed by y=2x−x^2 and the x-axis (oriented positively).

Answer 1

∫_CF • n d s = ∫∫_D div F d A = ∫∫_D [ ∂ P /∂ x + ∂ Q / ∂ y ] d A = ∫∫_D [ 1/2√x + 5 } d A

∫²₀ ∫^{2x-x^2}₀ [ 1/(2√x) + 5] d y d x = ∫²₀ [ y / (2√x) - 5 y ] |^{2x-x^2}₀ d x = ∫²₀ [ (2x-x²) / (2√x) - (2x-x²) ] d x=

∫²₀ [ √x - (1/2) x^3/2 -2x + x² ] d x = 4 √2 /3 - 7/ 3.