evaluate ⃗⃗ F · d⃗r, where F(x, y) = (x + 2y, 2x + y) and C is the line segment from

Question

Using two different methods, evaluate ⃗⃗

F · d⃗r, where F(x, y) = (x + 2y, 2x + y) and C is the line segment from

(0, 0) to (1, 1). Both methods should give you the same answer!

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Answer 1

1) Let C is line segment combining (0,0) and (1,1). equation is y = x and 0 ≤ x ≤ 1. dy = dx

Then ∫(C)F(x, y)dr = ∫₀¹((x + 2x) + (2x +x))dx = ∫₀¹6xdx = 3x₀¹ = 3.

2) Because ∂(2x + y)/∂x = ∂(x + 2y)/∂y = 2 then there is a function f(x,y) such that df = (x + 2y)dx + (2x + y)dy.∂

∂f/∂x = x + 2y, f(x,y) = ∫(x + 2y)dx = x²/2 + 2yx + g(y), then ∂f/dy = 2x + g'(y) = 2x + y, or g'(y) = y, g(y) = ∫ydy = y²/2 + C. Then f(x,y) = x²/2 + 2xy + y²/2 + C.

Then ∫(C)F(x, y)dr = f(1,1) - f(0,0) = (1/2 + 2 + 1/2 + C) - (0 + 0 + 0 +C) = 3.

We use fundamental theorem of calculus for line integral.

Answer: 3