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\title[Problem Set 9]{Problem Set 9.  Due Monday, 18 September}

\begin{document}

\maketitle



\noindent {\bf Reading.} {\em Quick Calculus}, pp. 199--207.

\noindent {\bf Supplementary reading.} Simmons, Chapters 18, 19, 20.

\begin{enumerate}

\item There is a tent that has a circular base of radius $1$ meter.
The tent consists of stretching nylon fabric over a vertical
semicircular pole (also of radius $1$ meter) attached to the base at
the opposite ends of a diameter.  What is the volume of the tent?
(Hint: The cross-sections are triangles.)

\item Rotate the circle $(x-4)^2+y^2=1$ around the $y$-axis.  The
solid that you get is a doughtnut, also called a {\em torus}.
Calculate the volume of this torus using the disc method.  (Your discs 
will look like washers.)

\item There is a $3$-dimensional solid with base in the $xy$-plane.
The base is triangular, with vertices $(0,0,0)$, $(2,2,0)$ and
$(2,-2,0)$.   The cross-sections of the solid when it is cut by planes
perpendicular to the $x$-axis are squares.  Compute the volume of this 
solid.

\item There is a $3$-dimensional solid with base in the $xy$-plane.
The base is triangular, with vertices $(0,0,0)$, $(2,2,0)$ and
$(2,-2,0)$.   The cross-sections of the solid when it is cut by planes
perpendicular to the $x$-axis are semi-circles.  Compute the volume of this 
solid.



\item Compute the following multiple integrals.

\begin{enumerate}
\item $\int_{y=0}^{1}\int_{x=0}^1\ x^2y+xy^2\ dx\ dy$
\item $\int_{y=0}^{1}\int_{x=0}^y\ 2\sqrt{y^2+1}\ dx\ dy$
\item $\int_{x=0}^{1}\int_{y=0}^{e^x}\ 4y\ dy\ dx$
\end{enumerate}



%\item Compute the following partial derivatives.
%
%\begin{enumerate}
%\item $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$
%of $f(x,y)=x+xy^2+3xy+x^2y+y$
%\item $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$,
%and $\frac{\partial}{\partial z}$ of $f(x,y,z)=xyz+xy-z$
%\item $\frac{\partial}{\partial x_i}$ ($i=1,\dots ,n$) of
%$f(x_1,x_2,\dots ,x_n)=x_1+x_2+\cdots +x_n$ 
%\end{enumerate}





\end{enumerate}





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