Question

Вычислить криволинейный интеграл

Answer 1

$x=t-sint\ ,\ y=1-cost\ \ ,\ \ 0\leq t\leq 2\pi \\\\\\\int\limits_{L}\, (x+y)\, dx+(x-y)\, dy=\\\\=\int\limits^{2\pi }_0\, (t-sint+1-cost)\cdot (1-cost)\, dt+(t-sint-1+cost)\cdot sint\, dt=\\\\=\int\limits^{2\pi }_0(t-sint+1-cost-t\, cost+sint\, cost-cost+cos^2t+t\, sint-sin^2t-sint+\\\\+sint\, cost)\, dt=\int\limits^{2\pi }_0(t-2sint+1-2cost-t\, cost+2\, sint\, cost+cos2t+t\, sint)\, dt=$

$=\Big(\dfrac{t^2}{2}+2cost+t-2sint-(t\, sint+cost)+sin^2t+\dfrac{1}{2}\, sin2t+(-t\, cost+sint)\Big)\Big|_0^{2\pi }=\\\\\\=2\pi ^2+2(1-1)+2\pi -0-(1-1)+0+0+(-2\pi )=2\pi ^2$