6.4 Green’s theorem  (Page 10/12)

Use Green’s theorem to evaluate line integral ${\displaystyle {\oint }_C{e}^{2x}\text{sin}2ydx+e^{2x}\text{cos}2ydy},$ where C is ellipse $9{(x-1)}^2+4{(y-3)}^2=36$ oriented counterclockwise.

Evaluate line integral ${\displaystyle {\oint }_C{y}^{2}dx+x^{2}dy,}$ where C is the boundary of a triangle with vertices $\left(0,0\right),\left(1,1\right),\text{and}\left(1,0\right),$ with the counterclockwise orientation.

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C\text{h}·d\text{r}}$ if $\text{h}\left(x,y\right)=e^y\text{i}-\text{sin}\pi x\text{j},$ where C is a triangle with vertices (1, 0), (0, 1), and $\left(\mathrm{-1},0\right)$ $\left(\mathrm{-1},0\right)$ traversed counterclockwise.

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}\sqrt{1+x^3}dx+2xydy}$ where C is a triangle with vertices (0, 0), (1, 0), and (1, 3) oriented clockwise.

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}{x}^{2}ydx-x{y}^{2}dy}$ where C is a circle $x^2+y^2=4$ oriented counterclockwise.

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}\left(3y-e^{\text{sin}x}\right)dx}+\left(7x+\sqrt{y^4+1}\right)dy$ where C is circle $x^2+y^2=9$ oriented in the counterclockwise direction.

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}(3x-5y)dx}+(x-6y)dy\text{,}$ where C is ellipse $\frac{x^2}{4}+y^2=1$ and is oriented in the counterclockwise direction.

Let C be a triangular closed curve from (0, 0) to (1, 0) to (1, 1) and finally back to (0, 0). Let $\text{F}(x,y)=4y\text{i}+6x^2\text{j}.$ Use Green’s theorem to evaluate ${\displaystyle {\oint }_C\text{F}·d\text{s}}.$

Use Green’s theorem to evaluate line integral ${\displaystyle {\oint }_{C}ydx-xdy}\text{,}$ where C is circle $x^2+y^2=a^2$ oriented in the clockwise direction.

Use Green’s theorem to evaluate line integral ${\displaystyle {\oint }_C(y+x)dx+(x+\text{sin}y)dy}\text{,}$ where C is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.

Use Green’s theorem to evaluate line integral ${\displaystyle {\oint }_C\left(y-\text{ln}\left(x^2+y^2\right)\right)dx+\left(2\text{arctan}\frac{y}{x}\right)dy},$ where C is the positively oriented circle ${\left(x-2\right)}^2+{\left(y-3\right)}^2=1.$

Use Green’s theorem to evaluate ${\displaystyle {\oint }_{C}xydx+x^3{y}^{3}dy,}$ where C is a triangle with vertices (0, 0), (1, 0), and (1, 2) with positive orientation.

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}\text{sin}ydx+x\text{cos}ydy},$ where C is ellipse $x^2+xy+y^2=1$ oriented in the counterclockwise direction.

Let $\text{F}(x,y)=\left(\text{cos}\left(x^5\right)\right)-\frac{1}{3}{y}^3\text{i}+\frac{1}{3}{x}^3\text{j}.$ Find the counterclockwise circulation ${\displaystyle {\oint }_C\text{F}·d\text{r}},$ where C is a curve consisting of the line segment joining $(\mathrm{-2},0)\text{and}(\mathrm{-1},0)\text{,}$ half circle $y=\sqrt{1-x^2},$ the line segment joining (1, 0) and (2, 0), and half circle $y=\sqrt{4-x^2}.$

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}\text{sin}\left(x^3\right)dx+2y{e}^{x^2}dy},$ where C is a triangular closed curve that connects the points (0, 0), (2, 2), and (0, 2) counterclockwise.

Let C be the boundary of square $0\le x\le \pi \text{,}0\le y\le \pi \text{,}$ traversed counterclockwise. Use Green’s theorem to find ${\displaystyle {\int }_C^{}\text{sin}(x+y)dx+\text{cos}(x+y)dy}.$

Use Green’s theorem to evaluate line integral ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where $\text{F}(x,y)=\left(y^2-x^2\right)\text{i}+\left(x^2+y^2\right)\text{j},$ and C is a triangle bounded by $y=0\text{,}x=3,\text{and}y=x\text{,}$ oriented counterclockwise.

Use Green’s Theorem to evaluate integral ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where $\text{F}(x,y)=\left(x{y}^2\right)\text{i}+x\text{j},$ and C is a unit circle oriented in the counterclockwise direction.

Use Green’s theorem in a plane to evaluate line integral ${\displaystyle {\oint }_C\left(xy+y^2\right)dx+x^{2}dy},$ where C is a closed curve of a region bounded by $y=x\text{and}y=x^2$ oriented in the counterclockwise direction.

Calculate the outward flux of $\text{F}=\text{−}x\text{i}+2y\text{j}$ over a square with corners $\left(\mathrm{\pm 1},\mathrm{\pm 1}\right),$ where the unit normal is outward pointing and oriented in the counterclockwise direction.

[T] Let C be circle $x^2+y^2=4$ oriented in the counterclockwise direction. Evaluate ${\displaystyle {\oint }_C\left[\left(3y-e^{\text{tan}-1_x}\right)dx+\left(7x+\sqrt{y^4+1}\right)dy\right]}$ using a computer algebra system.

Find the flux of field $\text{F}=\text{−}x\text{i}+y\text{j}$ across $x^2+y^2=16$ oriented in the counterclockwise direction.

Let $\text{F}=\left(y^2-x^2\right)\text{i}+\left(x^2+y^2\right)\text{j},$ and let C be a triangle bounded by $y=0,x=3,$ and $y=x$ oriented in the counterclockwise direction. Find the outward flux of F through C .

[T] Let C be unit circle $x^2+y^2=1$ traversed once counterclockwise. Evaluate ${\displaystyle {\int }_C\left[\text{−}{y}^3+\text{sin}\left(xy\right)+xy\text{cos}\left(xy\right)\right]dx+\left[x^3+x^2\text{cos}\left(xy\right)\right]dy}$ by using a computer algebra system.

[T] Find the outward flux of vector field $\text{F}=x{y}^2\text{i}+x^{2}y\text{j}$ across the boundary of annulus $R=\left\{\left(x,y\right):1\le x^2+y^2\le 4\right\}=\left\{\left(r,\theta \right):1\le r\le 2,0\le \theta \le 2\pi \right\}$ using a computer algebra system.

Consider region R bounded by parabolas $y=x^2\text{and}x=y^2.$ Let C be the boundary of R oriented counterclockwise. Use Green’s theorem to evaluate ${\displaystyle {\oint }_C\left(y+e^{\sqrt{x}}\right)dx+\left(2x+\text{cos}\left(y^2\right)\right)dy.}$