Calculus Green theorem

Question

If C is a circle of radius 7 centered at the origin, then evaluate ∮C(4y−^esin(x))dx+(11x−sin(y^3+y))dy.

Answer 1

∫_C [4 y - e^sinx ] d x + [ 11x - sin ( y³ +y ) ] d y ., where C : x² + y² = 49

P (x, y) = 4 y - e^sinx Q(x, y ) = 11x - sin ( y³ +y)

∂ Q/ ∂ x = ∂/∂x [ 11x - sin ( y³ +y) ] = 11

∂ P/ ∂ x = ∂/∂y [ 4 y - e^sinx ] = 4 Then

∫_C [4 y - e^sinx ] d x + [ 11x - sin ( y³ +y ) ] d y = ∫∫_D [ ∂ Q/ ∂ x - ∂ P/ ∂ x ] d A , where D : x² + y² ≤ 49

∫∫_D [ ∂ Q/ ∂ x - ∂ P/ ∂ x ] d A = ∫∫_D [ 7 ] d A = ∫₀^2π ∫₀⁷ 7 r dr dθ =

7 ∫₀^2π ∫₀⁷ r dr dθ = ( 7 / 2 ) ∫₀^2π [ r²]|₀⁷d θ =(343/ 2) ∫₀^2πd θ = (343/ 2) 2π =343 π