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\title{\vspace*{-1.5in}\textbf{\S6.6 Part II: Center of Mass}}
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\begin{document}

\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{February 19, 2018}
\chead{Center of Mass}
\vspace{-.5in}
\noindent
\textbf{Key Points:}
\begin{itemize} 
\item The \textbf{center of mass} (or centroid) of a thin plate is:\vskip1cm
\item For a system of $n$ particles with masses $m_1, \ldots, m_n$ located at the points $(x_1,y_1), \ldots, (x_n,y_n)$ in the $xy$-plane, the center of mass of the system is located at:\vskip 1in
\item The \textbf{moment of the system about the y-axis} is
\[
M_y = \phantom{alsdkfjalsdkfjalskdjflaksdjflaksdjflkasj}
\]
This measures \rule{4in}{.75pt}.
\item The \textbf{moment of the system about the x-axis} is
\[
M_x = \phantom{alsdkfjalsdkfjalskdjflaksdjflaksdjflkasj}
\]
This measures \rule{4in}{.75pt}.
\item In the case where we are looking at a thin region bounded by the curves $y=f(x)$ and $y=g(x)$, we chop the region in to small rectangles that we consider to be point masses. In this case:\newpage
\end{itemize}
Examples:
\vskip12pt
\begin{enumerate}
\item Find the moments $M_x$ and $M_y$ and the center of mass of the system of the following point masses:
\begin{itemize}
\item A mass of 6 at the point (1,5)
\item A mass of 5 at the point (3,-2)
\item A mass of 10 at the point (-2,-1)
\end{itemize}\vfill
\item Find the centroid of the region bounded by the curves $y=\sqrt{x}$ and $y=x$.\vfill
\newpage
\item Find the center of mass of the semicircular plate of radius $r$.\vfill
\item Find the center of mass of the region between the $x$-axis and the parabola $y=x^2+1$ between $x=-2$ and $x=2$.\vfill
\end{enumerate}
\end{document}
