Evaluate the integral C ⃗⃗ F · d⃗r, where F(x, y) = (y cos x − xy sin x, xy + x cos x) and C is the triangle consisting of line segments from

Question

Evaluate the integral C ⃗⃗

F · d⃗r, where F(x, y) = (y cos x − xy sin x, xy + x cos x) and C is the triangle consisting of line segments from

(0,0) to (3,0), from (3,0) to (0,6), and from (0,6) to (0,0).

Answer 1

We will use Green's thorem: ∫(C)(ycosx - xysinx)dx + (xy + xcosx)dy = ∫∫(R)(∂( xy + xcosx)/∂x - ∂(ycosx -xysinx)/∂y)dxdy = ∫∫(R)(y + cosx - xsinx - cox + xsinx)dxdy = ∫∫(R)ydxdy = ∫₀³dx∫₀^{-2x + 6}ydy = ∫₀³y²/2₀^{-2x + 6}dx = ∫₀³(- 2x + 6)²/2dx = 1/2 ∫₀³(4x² - 24x + 36)dx = 1/2(4x³/3 - 12x² + 36x)₀³ = 18 - 54 + 54 = 18.