Exercise $9.$ Use Green's theorem to evaluate the line integral

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(e^{x^3}-xy)dx+(x^2+ln(1+y^y))dy$

where $C$ is the square with sides $x=0,x=1,y=0$ and $y=1.$ Assume the counterclockwise orientation for $C.$

Exercise $10.$ Use Green's theorem to find the area bounded by the curve $C$ with parametric equations:

$r(t)=(2-3cost)i+(5+2sint)j,0t2$

Exercise $11.$ Evaluate the surface integral

${}_S(x+y+z)dS$

where $S$ is the surface with parametric equations $x=u+v,y=u-v,z=1-u$ where $0u1,0v1.$

Exercise $12.$ Evaluate the surface integral

${}_{S}F*dS$

where $F=+yj+(5-3z)k$ and $S$ is the part of the paraboloid $z=1-x^2-y^2$ lying above the plane $z=0,$ and $S$ has upward orientation.

Exercise $13.$ Find the area under the graph of $z=100(x^2+2y^2)$ lying above the second quadrant portion of the curve $x^2+y^2=4.$

Exercise $14.$ I am doing laps around the unit circle $($counterclockwise$)$ in the presence of the force field

$F=($:Axy$-B{y}^3,4y+3x^2-3x{y}^2$:$).$

After having gone from $(1,0)$ to $(0,1),$ I am already getting tired from all of the work I've done. A friend standing nearby tells me to chill because when I get back to $(1,0)$ I will have done no work at all. What are A and $B?$ Briefly explain.

If I go around the circle too much, I'll get dizzy so my friend tells me to go from $(-1,0)$ to $(3,-2)$ along the path $y=\sqrt[\phantom{2}]{x+1}(x-2{)}^{300}(x-4{)}^{301}$ instead. How much work will I do walking on that path?

Exercise $15.$ Use Green's Theorem to evaluate

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}-y^{3}dx+(x^3+\sqrt[\phantom{2}]{y^3+1})dy$

where $C$ is the closed counterclockwise path consisting of the bottom half of the unit circle and the portion of the $x-$axis with $-1x1.$

SOLVE ALL FOR CALCLUS$3$