그린 정리

${\int }_{C}^{}2xydx+(x+y)dy,$ where C is the boundary of the region lying between the graphs of $y=0$ and $y=4-{x}^{2}$ oriented in the counterclockwise direction

${\int }_{C}^{}2\,\arctan \left(\frac{y}{x}\right)dx+\ln ({x}^{2}+{y}^{2})dy,$ where C is defined by $x=4+2\,\cos \,\theta ,y=4\,\sin \,\theta$ oriented in the counterclockwise direction

${\int }_{C}^{}\sin \,x\,\cos \,ydx+(xy+\cos \,x\,\sin \,y)dy,$ where C is the boundary of the region lying between the graphs of $y=x$ and $y=\sqrt{x}$ oriented in the counterclockwise direction

${\int }_{C}^{}xydx+(x+y)dy,$ where C is the boundary of the region lying between the graphs of ${x}^{2}+{y}^{2}=1$ and ${x}^{2}+{y}^{2}=9$ oriented in the counterclockwise direction

${\int }_{C}(-ydx+xdy),$ where C consists of line segment C₁ from $(-1,0)$ to (1, 0), followed by the semicircular arc C₂ from (1, 0) back to (–1, 0)

Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0). Find the value of ${\int }_{C}xydx+\sqrt{{y}^{2}+1}dy.$

Evaluate line integral ${\int }_{C}x{e}^{-2x}dx+({x}^{4}+2{x}^{2}{y}^{2})dy,$ where C is the boundary of the region between circles ${x}^{2}+{y}^{2}=1$ and ${x}^{2}+{y}^{2}=4,$ and is a positively oriented curve.

Evaluate ${\int }_{C}{y}^{3}dx-{x}^{3}{y}^{2}dy,$ where C is the positively oriented circle of radius 2 centered at the origin.

Calculate ${\int }_{C}-{x}^{2}ydx+x{y}^{2}dy,$ where C is a circle of radius 2 centered at the origin and oriented in the counterclockwise direction.

Calculate integral ${\int }_{C}2[y+x\,\sin (y)]dx+[{x}^{2}\cos (y)-3{y}^{2}]dy$ along triangle C with vertices (0, 0), (1, 0) and (1, 1), oriented counterclockwise, using Green’s theorem.

For the following exercises, use Green’s theorem to find the area.

Find the area between ellipse $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$ and circle ${x}^{2}+{y}^{2}=25.$

Use Green’s theorem to evaluate ${\int }_{{C}^{+}}({y}^{2}+{x}^{3})dx+{x}^{4}dy,$ where ${C}^{+}$ is the perimeter of square $[0,1]\,\times \,[0,1]$ oriented counterclockwise.

Use Green’s theorem to find the area under one arch of the cycloid given by parametric plane $x=t-\sin \,t,y=1-\cos \,t,t\ge 0.$

[T] Verify Green’s theorem by using a computer algebra system to evaluate the integral ${\int }_{C}x{e}^{y}dx+{e}^{x}dy,$ where C is the circle given by ${x}^{2}+{y}^{2}=4$ and is oriented in the counterclockwise direction.

Evaluate ${\int }_{C}({x}^{2}y-2xy+{y}^{2})ds,$ where C is the boundary of the unit square $0\le x\le 1,\,0\le y\le 1,$ traversed counterclockwise. Use $ds = dx + dy$ for this exercise.

Evaluate ${\int }_{C}^{}\frac{-(y+2)dx+(x-1)dy}{{(x-1)}^{2}+{(y+2)}^{2}},$ where C is any simple closed curve with an interior that does not contain point $(1,-2)$ traversed counterclockwise.

Evaluate ${\int }_{C}^{}\frac{xdx+ydy}{{x}^{2}+{y}^{2}},$ where C is any piecewise, smooth simple closed curve enclosing the origin, traversed counterclockwise.

For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C.

$\text{F}(x,y)=xy\text{i}+(x+y)\text{j},$ $C:{x}^{2}+{y}^{2}=4$

For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C.

$\text{F}(x,y)=({x}^{3/2}-3y)\text{i}+\left(6x+5\sqrt{y}\right)\text{j},$ C : boundary of a triangle with vertices (0, 0), (5, 0), and (0, 5)

Evaluate ${\int }_{C}(2{x}^{3}-{y}^{3})dx+({x}^{3}+{y}^{3})dy,$ where C is a unit circle oriented in the counterclockwise direction.