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\noindent {\bf Math 222 HW\#4, due Sunday 3/7/21} \hspace{.5in} \textbf{NAME:} \hspace{4in} \\

%outlier malcom
\noindent If you do the Folland Zero Content Proof (\#1) you need not do \#2 \& \#5 and you will still receive full credit.  Alternatively, if you do \#2-6, you need not do the Folland Zero Content proof (\#1) and you will still receive full credit.
\begin{enumerate}
\item Folland Proof of Theorem 4.13 (Technical refinement of Theorem 4.12) \\
Prove that if $f$ is bounded on $[a,b]$ and the set of points in $[a,b]$ at which $f$ is discontinuous has zero content, then $f$ is integrable on $[a,b]$.


%\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 Problem 8-3: Jones\\
%For the record, here is the general principle used above.  Prove that for a general real symmetric $n\times n$ matrix $A$ which is positive semi-definite and for $t\in \R^n$, 
%\[
%At \cdot t = 0 \mbox{ if and only if } At=0.
%\]
%(It will be easiest to do each direction separately.)

\item Stewart\\
Use polar coordinates to combine the sum
\[
\int_{\frac{1}{\sqrt{2}}}^1 \int_{\sqrt{1-x^2}}^x xy \ dy dx + \int_1^{\sqrt{2}}\int_0^x xy \ dy dx +  \int_{\sqrt{2}}^2\int_0^{\sqrt{4-x^2}} xy \ dy dx
\]
into one double integral.  Then evaluate the double integral.

\item Stewart \& Jones 9.G 
\begin{enumerate}
\item We define the improper integral (over the entire plane $\R^2$)
\[
I = \iint_{\R^2}e^{-(x^2+y^2)} \ dA = \int_{-\infty}^\infty \int_{-\infty}^\infty  e^{-(x^2+y^2)} \ dy dx = \lim_{a \to \infty} \iint_{D_a} e^{-(x^2+y^2)} \ dA 
\]
where $D_a$ is the disk with radius $a$ and center the origin.  Show that
\[
\int_{-\infty}^\infty  \int_{-\infty}^\infty e^{-(x^2+y^2)} \ dA = \pi
\]
\item An equivalent definition of the improper integral in part (a) is
\[
\iint_{\R^2}e^{-(x^2+y^2)} \ dA = \lim_{a \to \infty} \iint_{S_a} e^{-(x^2+y^2)} \ dA 
\]
where $S_a$ is the square with vertices $(\pm a, \pm a)$.  Use this to show that 
\[
\int_{-\infty}^\infty e^{-x^2} dx \int_{-\infty}^\infty e^{-y^2} dy  = \pi
\]
\item Deduce that
\[
\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi} 
\]
\item By making the change of variable $t = x\sqrt{2}$, show that
\[
\int_{-\infty}^\infty e^{-x^2/2} dx = \sqrt{2\pi} 
\]
(This is a fundamental result for probability and statistics).
\end{enumerate}

\item Stewart \& Jones 9.G 
Use the result of the previous exercise, part (c) to evaluate:
\begin{enumerate}
\item $\displaystyle \int_0^\infty x^2 e^{-x^2} \ dx $
\item $\displaystyle \int_0^\infty \sqrt{x} e^{-x} \ dx $
\end{enumerate}

\item Stewart \& Jones  \\
Use a triple integral to find the volume of the given solid.  Also provide a sketch of the solid.
\begin{enumerate}
\item The tetrahedron enclosed by the coordinate planes and the plane $2x+y+z = 4$
\item The solid enclosed by the paraboloids $y=x^2 + z^2$ and $y=8-x^2-z^2$.
\end{enumerate}

\item Stewart \& Jones\\
Sketch the solid whose volume is given by the iterated integral. DO NOT EVALUATE!
\begin{enumerate}
\item $\displaystyle \int_0^1 \int_0^{1-x} \int_0^{2-2z} dy dz dx$
\item $\displaystyle \int_0^2 \int_0^{2-y} \int_0^{4-y^2} dx dz dy$

\end{enumerate}
%\item Problem 8-11: Jones \\
%Conclude immediately from the properties and summary on 8-14 and 8-15 that 
%\[
%{e}_1 \times \dots \times \check{e}_i \times \dots \times {e}_n = (-1)^{n+i}e_i,
%\]
%where the notation $\check{e}_i$ indicates that the vector $e_i$ has been omitted from the left hand side of the above expression.  Also we have omitted the decorations on all the other $e_j$, since this is more typical of notation in differential geometry and analysis classes.  (In these classes we use a $\hat{\ }$ to indicate omission of a vector.)

%\item Problem 8-12: Jones \\
%Next prove that
%\[
%({x}_1 \times  \dots \times {x}_{n-1}) \cdot ( {e}_1 \times \dots \times \check{e}_i \times \dots \times {e}_n ) = \det(x_1 \cdots \check{x}_i \cdots x_{n-1}),
%\]
%where the notation $\check{e}_i$ indicates that the vector $e_i$ has been omitted from the left hand side and the notation $\check{x}_i$ indicates that the $i$-th row $x_i$ has been omitted from the right hand side.
 



\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you find to provide a worthwhile learning experience?  Should I be assigning a similar number of problems, fewer problems, or more problems in the future?  Is there a good mix of theory and computations?

\vfill
\eject

\end{enumerate}

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