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{\bf HW 6\\Math 261, S19}

Please see the course syllabus for details on how to turn in your homework assignments.  This one is due at the beginning of your class on \underbar{\bf Friday,  March 29}.


\begin{enumerate}

\item (5 pts.) \textsc{True or False:}
\begin{enumerate}
\item Let $R$ denote a region in the $xy$-plane. The expression $\int_R dxdy$ computes the area of $R$.
\item Let $R$ denote a region in the $xy$-plane, and $f(x,y)$ a function from $\bR^2$ to $\bR$. The expression  $\int_R f(x,y)dxdy$ is always positive. 
\item Let $R$ denote a region in the $xy$-plane, and $f(x,y)$ a function from $\bR^2$ to $\bR$ such that $f(x,y)$ is non-negative for any point in $R$. The expression  $\int_R f(x,y)dxdy$  computes the volume under the graph of $f$ (and bounded below by the $xy$-plane).
\item  $\int_2^4\int_3^7dxdy=\int_3^7\int_2^4dydx$. 
\item  $\int_2^4\int_3^7 f(x,y) dxdy=\int_3^7\int_2^4f(x,y) dydx$ for any function $f(x,y)$. 
\end{enumerate}

\item(3 pts.) Suppose we wish to integrate $f(x,y) = 3x^2-xy+3$ over the rectangle given by $0\leq x\leq 2$, $1\leq y\leq 5$.  Set up this integral using the variable order $dxdy$ but do not compute the answer.

\item (3 pts.)  Now compute the answer to the previous problem.

\item (3 pts.)  Repeat problem 2 using variable order $dydx$ and compute the answer.

\item (3 pts.)  Suppose we wish to integrate some function $g(x,y)$ over the triangle with vertices $(0,0)$, $(0,2)$, and $(1,2)$.   Set up this integral using the variable order $dxdy$.  (Do not compute the answer.)

\item (3 pts.)  Compute the volume of the octahedron in $\mathbb{R}^3$ with vertices at $(0,0, \pm 1), (0,\pm1, 0), (\pm1, 0,0)$. (Hint: exploit the symmetries of the problem to reduce to a problem that may be computed as a simple double integral).

 
 

\end{enumerate}

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