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\begin{center}{ \Large{Worksheets}  \\\large{10/26/2017} }\end{center}
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\item Evaluate the following line integrals $\int_{C}{f(x,y,z)ds}$, where
\begin{enumerate}
\item $f(x,y,z)=x+y+z$ and $C(t)=(\sin t,\cos t,t)$, $t\in[0,2\pi]$\vspace{3.5in}
\item $f(x,y,z)=yz$ and $C(t)=(t,3t,2t)$, $t\in[1,3]$\vspace{3.5in}
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\newpage
\item Find an parametrization of the ellipse
\[\frac{{{(x-2)}^{2}}}{4}+\frac{{{(y-3)}^{2}}}{9}=1\]oriented counterclockwise. Then express the length of the ellipse as an integral (but don't evaluate).\vspace{4in}
\item Evaluate the following line integrals:
\begin{enumerate}
\item $\int_{C}{xdy-ydx}$ where $C(t)=(\cos t,\sin t)$, $t\in[0,2\pi]$\newpage
\item $\int_{C}{dy+dx}$ where $C$ is the ellipse given in Problem 2.\vspace{3.2in}
\end{enumerate}
\item Consider the force field $F(x,y,z)=\langle x,y,z\rangle$. Compute the work done in moving a particle along the parabola $y=x^2$, $z=0$ from $x=-1$ to $x=2$.\vspace{3.5in}
\item A wire has parametrization $c(t)=(t^2,t,3)$, $0\le t\le 1$. The mass density at point $(x,y,z)$ is given by $\rho(x,y,z)=y$. Compute the mass of the wire.
\newpage
Recall the Green's theorem
\[\int_{\partial D}{Pdx+Qdy}=\iint_{D}{\left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right)dA}\]
where $\partial D$ denotes the boundary of $D$, positively oriented.
\item Let $D$ be a region on the plane. Choose appropriate $P$ and $Q$ so that
\[\textrm{Area}(D)=\iint_{D}{1dA}=\int_{\partial D}{Pdx+Qdy}\]\vspace{3.5in}
\item Use the formula in Problem 6 to compute the area of the ellipse in Problem 2.
\vspace{3.8in}

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