그린 정리

${\int }_{C}^{}2xydx+(x+y)dy,$ where C is the path from (0, 0) to (1, 1) along the graph of $y={x}^{3}$ and from (1, 1) to (0, 0) along the graph of $y=x$ oriented in the counterclockwise direction

${\int }_{C}^{}2xydx+(x+y)dy,$ 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.

Find the counterclockwise circulation of field $\text{F}(x,y)=xy\text{i}+{y}^{2}\text{j}$ around and over the boundary of the region enclosed by curves $y={x}^{2}$ and $y=x$ in the first quadrant and oriented in the counterclockwise direction.

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.

Evaluate ${\int }_{C}{y}^{3}dx-{x}^{3}dy,$ where C includes the two circles of radius 2 and radius 1 centered at the origin, both with positive orientation.

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.

Evaluate integral ${\int }_{C}({x}^{2}+{y}^{2})dx+2xydy,$ where C is the curve that follows parabola $y={x}^{2}\,\text{from}\,(0,0)\text{ to }(2,4),$ then the line from (2, 4) to (2, 0), and finally the line from (2, 0) to (0, 0).

Evaluate line integral ${\int }_{C}(y-\sin (y)\cos (y))dx+2x\,{\sin }^{2}(y)dy,$ where C is oriented in a counterclockwise path around the region bounded by $x=-1,x=2,y=4-{x}^{2},$ and $y=x-2.$

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 ${x}^{2}+{y}^{2}=25.$

Find the area of the region enclosed by parametric equation

$$p(\theta )=(\cos (\theta )-{\cos }^{2}(\theta ))\text{i}+(\sin (\theta )-\cos (\theta )\sin (\theta ))\text{j}\,\text{for}\,0\le \theta \le 2\pi .$$

Find the area of the region bounded by hypocycloid $\text{r}(t)={\cos }^{3}(t)\text{i}+{\sin }^{3}(t)\text{j}.$ The curve is parameterized by $t\in [0,2\pi ].$

Find the area of a pentagon with vertices $(0,4),(4,1),(3,0),(-1,-1),$ and $(-2,2).$

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.