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\title{Volumes by slicing}
\date{September 3, 2020}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\mtd{
To calculate the volume of a solid (using the areas of slices):
\begin{enumerate}
\item Sketch the solid and a typical cross-section
\item Find a formula for $A(x)$
\item Find the limits of integration
\item Evaluate the integral $\displaystyle V = \int_a^b A(x) dx$
\end{enumerate}
}

\prob{
Find the volume of a solid whose base is the triangle with vertices $(0,0)$, $(0,1)$ and $(2,0)$ and with cross-sections perpendicular to to the $x$-axis are\dots
\begin{enumerate}[a)]
\item squares 
\item semi-circles
\end{enumerate}
}

\chal{
A water tank has the shape of a sphere with a diameter of 10ft.
If the depth of the water in the tank is 7ft, what percent of the tank is empty?
}

\newpage

\mtd{[Washer method]
To calculate the volume of a solid of rotation, use cross-sections perpendicular to the axis of rotation and $\displaystyle A(x) = \pi\left[ \left(R(x)\right)^2 - \left(r(x)\right)^2 \right]$ where $R(x)$ is the outer radius and $r(x)$ is the inner radius.
}

\prob{
Consider the region enclosed between the parabola $y=x^2$ and the line $y=1$.
Calculate the volume of the solid formed by rotating the region around \dots
\begin{enumerate}[a)]
\item \dots the line $y=1$
\item \dots the line $y=2$
\item \dots the line $x=1$
\end{enumerate}
}

\end{document}