벡터 미적분 전체 문제

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.

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 $r=3\,\sin \,2\theta .$ (Hint: $xdy-ydx={r}^{2}d\theta ).$

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

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

$$\text{r}(t)={t}^{2}\text{i}+\left(\frac{{t}^{3}}{3}-t\right)\text{j},\,-\sqrt{3}\le t\le \sqrt{3}.$$

[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.

A particle starts at point $(-2,0),$ 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 $\text{F}(x,y)=x\text{i}+({x}^{3}+3x{y}^{2})\text{j}.$

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 $(x,y)$ $(x,y)$ $(x,y)$ is $\text{F}(x,y)=({x}^{2}y+10y)\text{i}+({x}^{3}+2x{y}^{2})\text{j}.$ 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 $4{x}^{2}+{y}^{2}=4.$

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