6.4 Green’s theorem  (Page 8/12)

${\displaystyle {\int }_C^{}2xydx+(x+y)dy},$ 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

${\displaystyle {\int }_C^{}2xydx+(x+y)dy},$ 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

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

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

${\displaystyle {\int }_C^{}xydx+(x+y)dy},$ 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

${\displaystyle {\oint }_C(\text{−}ydx+xdy)},$ where C consists of line segment C ₁ from $\left(\mathrm{-1},0\right)$ to (1, 0), followed by the semicircular arc C ₂ 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 ${\displaystyle {\int }_{C}xydx+\sqrt{y^2+1}dy.}$

Evaluate line integral ${\displaystyle {\int }_{C}x{e}^{\mathrm{-2}x}dx+\left(x^4+2x^2{y}^2\right)dy,}$ 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 $\text{F}\left(x,y\right)=xy\text{i}+y^2\text{j}$ 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 ${\displaystyle {\oint }_C{y}^{3}dx-x^3{y}^{2}dy},$ where C is the positively oriented circle of radius 2 centered at the origin.

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

Calculate ${\displaystyle {\oint }_C\text{−}{x}^{2}ydx+x{y}^{2}dy},$ where C is a circle of radius 2 centered at the origin and oriented in the counterclockwise direction.

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