6.4 Green’s theorem  (Page 9/12)

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

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

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

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

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

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

Use Green’s theorem to prove the area of a disk with radius a is $A=\pi a^2.$

Use Green’s theorem to find the area of one loop of a four-leaf rose $r=3\text{sin}2\theta .$ ( Hint : $xdy-ydx={\text{r}}^{2}d\theta ).$

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

Use Green’s theorem to find the area of the region enclosed by curve

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

Evaluate ${\displaystyle {\int }_C\left(x^{2}y-2xy+y^2\right)ds,}$ where C is the boundary of the unit square $0\le x\le 1\text{,}0\le y\le 1\text{,}$ traversed counterclockwise.

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

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

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

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

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

A particle starts at point $(\mathrm{-2},0),$ moves along the x -axis to (2, 0), and then travels along semicircle $y=\sqrt{4-x^2}$ to the starting point. Use Green’s theorem to find the work done on this particle by force field $\text{F}(x,y)=x\text{i}+\left(x^3+3x{y}^2\right)\text{j}.$

David and Sandra are skating on a frictionless pond in the wind. David skates on the inside, going along a circle of radius 2 in a counterclockwise direction. Sandra skates once around a circle of radius 3, also in the counterclockwise direction. Suppose the force of the wind at point $\left(x,y\right)$ $\left(x,y\right)$ $\left(x,y\right)$ is $\text{F}(x,y)=\left(x^{2}y+10y\right)\text{i}+\left(x^3+2x{y}^2\right)\text{j}.$ Use Green’s theorem to determine who does more work.

Use Green’s theorem to find the work done by force field $\text{F}(x,y)=(3y-4x)\text{i}+(4x-y)\text{j}$ when an object moves once counterclockwise around ellipse $4x^2+y^2=4.$