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\begin{document}
\title{Mathematics 226, Spring 2000}
\author{Midterm II}
\date{March 31, 2000}
\maketitle

\begin{enumerate}

\item Evaluate the following integrals:
  \begin{enumerate}
  \item $\displaystyle \int_0^\pi \int_0^1 x \, \sin(xy) \, dy \, dx$
        %$\displaystyle \int_0^\pi \int_0^1 x \, \cos(xy) \, dy \, dx$
  \item $\displaystyle \int_0^1 \int_x^1 e^{\tfrac{x}{y}} \, dy \, dx$
        %$\displaystyle \int_0^1 \int_y^1 e^{\tfrac{y}{x}} \, dx \, dy$   
  \end{enumerate}
\hfill(10+10)


\item
\begin{enumerate}
\item
Find the volume of the solid bounded by the $x$-$y$ plane and the
paraboloid with equation
$$ z = 1 - (x^2 + y^2)^2 . $$
%Find the volume of the solid bounded by the plane $z=1$ and the
%paraboloid with equation
%$$ z = (x^2 + y^2)^2 . $$

\item Find the total mass of this solid if the density function is
given by
$$ \rho = \frac1{\sqrt{x^2+y^2}} . $$
\end{enumerate}
\hfill(10+10)


\item Use the transformation
$x=u^2$, $y=v^2$
%$x=\tfrac12 u^2$, $y=\tfrac12 v^2$
to compute the area of the region bounded by the $x$-axis,
the $y$-axis, and the
graph of
$$ \sqrt x + \sqrt y = 1. $$
\hfill(20)


\item Find all local minimum and maximum values of the function
$$ f(x,y) = \frac1{x(x+6) + y(y-6)} .$$
%$$ f(x,y) = \frac1{x(x+4) + y(y-4)} .$$
\hfill(20)


\item Use the method of Lagrange multipliers to show that the
rectangle with maximum area that has a given perimeter $p$ is a
square.
\hfill(20)

\end{enumerate}
\end{document}
