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\begin{center}
{\Large Take home 3

Math 2210 - 2}
\end{center}

Instructions:
\begin{itemize}
\item  Due on Wednesday, April 11, at the beginning of class.
\item  There is no reason for sloppy or missing work.  I will take off points for sloppy or missing work.
\item  Show all your work.  If I can't follow your work or it is missing you WILL NOT get credit.
\item  A right answer without the right work and right explanation is not worth full credit!
\item  You must work alone, but you are free to use your book and calculator.
\item  I must be able to follow your work as if you had no calculator or book available.
\item  You are free to ask me questions.  It is possible (likely?) that I will not answer, but you can always ask.
\item  Drawing pictures where applicable to help me understand your work would be nice.
\end{itemize}

\hrule

\begin{enumerate}[1.]

\item  Recall that the average value of a function of one variable on the interval $[a,b]$ is defined by
	\[
	f_{ave} = \frac{\int_a^b f(x) \; dx}{b-a}
	\]
	Similarly, we can define average value for functions of two variables on a bounded set $D$.
	If $f(x,y)$ is a function of two variables, we define
	\[
	f_{ave} = \frac{\iint_D f(x,y) \; dA}{\iint_D dA}
	\]
	If you recall the geometric interpretation of average value for a function of one variable,
	you should be able to figure out the geometric interpretation for the average value of 
	a function of two variables.
\begin{enumerate}[(a)]
\item  Find the average value of $f(x,y)=e^{x+y}$ over the triangle with vertices $(0,0), (0,1), (1,0)$.
\item  How would you define average value for a function of three variables?
\item  Find the average value of $f(x,y,z)=2x+3y+z$ over the domain $D$ defined by
	\[
	1 \leq x \leq 2 \qquad
	-2 \leq y \leq 1 \qquad
	0 \leq z \leq 1
	\]
\end{enumerate}



\item  Consider the solid cylinder $x^2+y^2 \leq 1$, $1 \leq z \leq 2$ with density $\delta = (x^2+y^2)z^2$.
\begin{enumerate}[(a)]
\item  Find the mass of the solid.
\item  Find the center of mass of the solid.
\end{enumerate}


\item  Let $E$ be the ellipsoid
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1
\]
where $a,b,c$ are all positive.
\begin{enumerate}[(a)]
\item  Find the volume of $E$.
\item  Find the integral
	\[
	\iiint_E \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \right) \; dV
	\]
\end{enumerate}
Hints: You might want to change variables twice.
First, change the variables $(x,y,z)$ into $(u,v,w)$ so that $u=x/a, v=y/b, w=z/c$.



\item  The cylinder $x^2+y^2=x$ divides the unit sphere ($x^2+y^2+z^2=1$) into two regions $S_1$ and $S_2$.
	$S_1$ is the region of the sphere inside the cylinder and 
	$S_2$ is the region of the sphere outside the cylinder.
\begin{enumerate}[(a)]
\item  Find the area of $S_1$.
\item  Find the area of $S_2$.
\end{enumerate}
Hint: Use as much elementary geometry and symmetry as possible.



\end{enumerate}

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