∫ L (x ^ 2 + y ^ 2) DX + (x ^ 2-y ^ 2) dy L is a line segment from a (0,0) to B (1,1) to C (2,0)

∫ L (x ^ 2 + y ^ 2) DX + (x ^ 2-y ^ 2) dy L is a line segment from a (0,0) to B (1,1) to C (2,0)

Common method:
L1：y = x、dy = dx
L2：y = 2 - x、dy = - dx
∫L (x² + y²) dx + (x² - y²) dy
= ∫(0→1) 2x² dx + ∫(1→2) [x² + (2 - x)² + (x² - (2 - x)²)(- 1)] dx
= ∫(0→1) 2x² dx + ∫(1→2) 2(x - 2)² dx
= 2/3 + 2/3
= 4/3
Green's formula:
Add the line segment n: y = 0, Dy = 0, counterclockwise, and make l a closed region D
P = x² + y²、P'y = 2y
Q = x² - y²、Q'x = 2x
∮L (x² + y²) dx + (x² - y²) dy
= ∫∫D (2x - 2y) dxdy
= 2∫(0→1) dy ∫(y→2 - y) (y - x) dx
= 4/3
∫N (x² + y²) dx + (x² - y²) dy = ∫(0→2) x² dx = 8/3
- I(L) + I(N) = ∮(L)