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{\bf Math 222 in class problems} \hspace{.5in} \textbf{Week: February 15, 2021} %outlier malcom
\begin{enumerate}


%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.

\item Stewart \\
Evaluate $\iint_D (x + 2y) dA$, where $D$ is the region bounded by $y=2x^2$ and $y=1+x^2$. \\ Answer: 32/5 %Answer: $\frac{32}{5}$.

\item Evaluate the integral
\[
\int \int_R \sin(y^2)dA
\]
where $R$ is the region $x \leq y \leq 1, \ 0\leq x \leq 1$. 

  
 \item John Hopkins Handout \\
Attempt to evaluate both ${\displaystyle \iint_R f(x,y) \ dydx}$ and ${\displaystyle \iint_R f(x,y) \ dxdy}$
over ${R = \{ (x,y) \in [0,2] \times [0,1] \}}$ with 
\[
{f(x,y) = \left\{ \begin{array}{ll} f(x,y) = \dfrac{xy(x^2-y^2)}{(x^2+y^2)^3} &\mbox{ for } (x,y) \neq (0,0) \\  f(0,0) = 0 & \\ \end{array} \right.}
\]
Answers: $dy dx$ yields $1/5$ while $dx dy$ yields $-1/20$!?! \\

Optional: demonstrate that the following limit DNE: \vspace{-.125in}
\[
\displaystyle \lim_{(x,y) \to (0,0)}\dfrac{xy(x^2-y^2)}{(x^2+y^2)^3}
\] 



\item HW \#3 problem 4 -  Folland\\
For each of the following regions $S \subset \R^2$, express the double integral $\int \int_S f \ dA$ in terms of iterated integrals in two different ways, e.g. find the limits of integration for $dA=dxdy$ and $dA=dydx$.  YOU DO NOT NEED TO EVALUATE THE INTEGRALS!
\begin{enumerate}
\item $S=\mbox{ the region in the left half plane between the curve } y=x^3 \mbox { and the line } y=4x.$
\item $S=\mbox{ the triangle with vertices } (0,0), \ (2,2), $ and $(3,1)$.
\item $S= \mbox{ the region between the parabolas } y=x^2 \mbox{ and } y=6-4x-x^2$.
\end{enumerate}

\end{enumerate}
 \vfill
\eject



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