求曲线积分∫L (x+y)＾2dx-(x＾2+y＾2)dy

围成D的三条直线分别是
AB：y = x/2 + 1/2
BC：y = - 3x + 11
CA：y = 4x - 3
假设L是逆时针方向行走的。用格林公式：
P = (x + y)²、P'y = 2(x + y) = 2x + 2y
Q = (- x² - y²)、Q'x = - 2x
Q'x - P'y = - 2x - 2x - 2y = - 4x - 2y = - 2(2x + y)
∮L (x + y)² dx - (x² + y²) dy
= - 2∫∫ (2x + y) dxdy
= - 2[∫(1→2) dx ∫(x/2 + 1/2→4x - 3) (2x + y) dy + ∫(2→3) dx ∫(x/2 + 1/2→- 3x + 11) (2x + y) dy]
= - 2[(7/8)∫(1→2) (17x² - 22x + 5) dx - (7/8)∫(2→3) (3x² + 14x - 69) dx]
= - 2[(7/8)(35/3) - (7/8)(- 15)]
= - 140/3

普通方法：
AB：y = x/2 + 1/2、dy = 1/2 dx、x：1→3
BC：y = - 3x + 11、dy = - 3 dx、x：3→2
CA：y = 4x - 3、dy = 4 dx、x：2→1
∫AB = ∫(1→3) {(x + x/2 + 1/2)² - [x² + (x/2 + 1/2)²](1/2)} dx
= ∫(1→3) (13x²/8 + 5x/4 + 1/8) dx
= 58/3
∫BC = ∫(3→2) {(x - 3x + 11)² - [x² + (- 3x + 11)²](- 3)} dx
= ∫(3→2) (34x² - 242x + 484) dx
= - 283/3
∫CA = ∫(2→1) {(x + 4x - 3)² - [x² + (4x - 3)²](4)} dx
= ∫(2→1) (- 43x² + 66x - 27) dx
= 85/3
所以∮L = ∫AB + ∫BC + ∫CA = - 140/3

逆时针是- 140/3
顺时针是140/3