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{\bf HW 7\\Math 261}

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, November 1}.


\begin{enumerate}
\item (5 pts.) \textsc{True or False:}
\begin{enumerate}
\item Let $R$ denote a region in the $xy$-plane,  given polar coordinates $(r, \theta)$. The expression $\int_R drd\theta$ computes the area of $R$.
\item Let $R$ denote a region in the $xy$-plane,  given polar coordinates $(r, \theta)$. The expression $\int_R rdrd\theta$ computes the area of $R$.
\item Let $R$ denote the plane region bounded by a circle of radius one centered at the origin. Then $\int_R x dx dy  = 0$.
\item Let $R$ denote the plane region bounded by a circle of radius one centered at the origin and $(r, \theta)$ polar coordinates. Then $\int_R r dr d\theta  = 0$.
\item If $(x,y)$ are cartesian coordinates and $(r, \theta)$ polar coordinates for the plane, and $R$ some plane region,  the expression
$$
\int_R f(x,y) dx dy = \int_R g(r, \theta) dr d\theta
$$
is true when 
$$
g(r, \theta) = r f(r\cos(\theta), r\sin (\theta) ).
$$
\end{enumerate}

\item (3 pts.) Set up but do {\bf NOT} evaluate a double integral to compute the integral of $f(x,y) = \cos(xy)$ over the part of the unit disk (the region inside the circle of radius 1 centered at the origin) in the first quadrant (where $x>0$, $y>0$).  
\item  (3 pts.) Convert the following double integral to an equivalent \underbar{polar} form but do {\bf NOT} evaluate:  $$\displaystyle \int_0^1\int_y^{\sqrt{4-y^2}} x^2+y^2 dxdy$$

\item  (3 pts.) Set up but do {\bf NOT} evaluate a triple integral to compute the volume of the tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,2,0)$, and $(0,0,1)$.   The top plane of the tetrahedron is given by $2x+y+2z=2$.   {\bf USE} the order $dz\ dy\ dx$.

\item  (3 pts.) Consider the tetrahedron $T$ with vertices $(1,0,0)$, $(1, -1, 1)$, $(1,1,1)$, and $(0,0,1)$.  How many regions must $T$ be split into in order to integrate some function over $T$ with the following variable orders (each worth 1 point)?  (Each answer is just 1 number!)
\begin{enumerate}
  \item $dx\ dy\ dz$
  \item $dx\ dz\ dy$
  \item $dy\ dz\ dx$
\end{enumerate}
(It would be good practice to try setting these integrals up, but that's not required for the problem.)
\item   (3 pts.) Let $R$ be the region colored in black in the figure below. The two curves bounding $R$ are the circle $x^2+y^2=1$ and the curve described in polar coordinates by the equation $r=2\sin(2\theta)$.  Set up but do \underbar{\bf NOT} evaluate a (sum of) double integral(s) in {\em polar} coordinates to find the area of $R$.

\includegraphics[scale=0.4]{clover.png}



\end{enumerate}

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