Using Greens Theorem

Question

Use Greens theorem to evaluate

∫ F • dr along the given positively oriented curve C where:

C

F= (xy − 4 ye^2x )i + ( y² − 2e^2x )j and C is the boundary of the region in the first quadrant determined by the graphs of y = 0, y = x²and x =1.

Need all the help I can get!

Answer 1

∫F•dr = ∫(xy - 4ye^2x)dx + (y² - 2e^2x)dy = ∫∫[∂/∂x(y² - 2e^2x) - ∂/∂y(xy - 4ye^2x)]dxdy = ∫∫(- 4e^2x - x + 4e^2x)dxdy =

C C D D

= ∫∫-xdxdy

D

D: 0 ≤ y ≤ x², 0 ≤ x ≤ 1.

So ∫∫-xdxdy = ∫₀¹-x∫₀^{x^2}dydx = ∫₀¹-xy₀^{x^2}dx = ∫₀¹(-x³)dx = -x⁴/4₀¹ = - 1/4. Answer: - 1/4

D