INTEGRAL CALCULUS

$1.)$ A vector field $F$ is given by $F(x,y)=(y{e}^x+siny)i+(e^x+xcosy)j.$ Determine whether or not $F$ is a conservative vector field. If it is$,$ find the potential function.

$2.)$ Let $F(x,y)=($:$y-ln(x^2+y^2),2arctan(\frac{y}{x})$:$)$ and $C$ be the circle $(x-2{)}^2+(y-3{)}^2=1$ oriented counterclockwise. Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr.$

$3.)$ Find the area of the surface which is the part of the paraboloid $x=y^2+z^2$ that lies inside the cylinder $y^2+z^2=9.$

$4.)$ Evaluate ${}_{S}F.dS$ where $F(x,y,z)=x^{4}i-x^3{z}^{2}j+4x{y}^{2}zk$ and $S$ is the surface of the solid bounded by the cylinder $x^2+y^2=1$ and the planes $z=x+2$ and $z=0.,$

$5.)$ Find the volume of the solid region above the surface $z=(x^2+y^2{)}^{\frac{1}{4}}$ and inside the sphere $x^2+y^2+z^2=2.$