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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 7}
\date{Due via Gradescope, Tuesday, April 14 before 20:00}
\maketitle

\emph{Note:} Assignments marked (*) will not be graded.  Do not turn
them in.  However, they will be discussed in the tutorial and example
solutions can be found in the appendix of Kantorovitz' book.  (All
Exercises are from Kantorovitz, Section 4.2.15.)

\begin{enumerate}

\item *Let $D \subset \R^2$ be the domain bounded by the parabola
$x=y^2$ and the line $x=y$.  Compute
\[
  \int_D \sin \frac{\pi x}y \, \d S \,.
\]

\item Let $D \subset \R^2$ be the annulus with radii $0<a<b$.  Compute
\[
  \int_D \arctan \frac{y}x \, \d S \,.
\]

\item Let $D \subset \R^2$ be the domain bounded by the parabola
$y=x^2$ and the lines $x=1$ and $y=0$.  Compute
\[
  \int_D y \, \e^{x^5} \, \d S \,.
\]

\item Let $R>0$.
\begin{enumerate}
\item Let $F \in C^1([0,R^2])$ and set $f = F'$.  Show that
\[
  \int_{B(0,R)} f(x^2 + y^2) \, \d S = \pi \, (F(R^2)-F(0)) \,.
\]
\item Calculate $\displaystyle \int_{B(0,R)} \cos(x^2+y^2) \, \d S$.
\item Calculate $\displaystyle \int_{B(0,R)} \exp(-x^2-y^2) \, \d S$.
\end{enumerate}

\item *Let
\[
  D = \{(x,y) \in \R^2 \colon x,y \geq 0; \sqrt x + \sqrt y \leq 1 \}
  \,.
\]
Compute
\[
  \int_D (\sqrt x + \sqrt y)^3 \, \d S \,.
\]
\emph{Hint:} use the map
\[
  (r, \phi) \mapsto (x,y) = r^4 \, (\cos^4 \phi, \sin^4 \phi) \,.
\]

\item Let
\[
  D = \{ (x,y) \in \R^2 \colon 1 \leq x^2 - y^2 \leq 9, 0 \leq x \leq
  4, y \geq 0 \} \,.
\]
Compute
\[
  \int_D x \, y \, \e^{x^2-y^2} \, \d S \,.
\]

\item *Let $0<a<b$ and 
\[
  D = \{ x \in \R^3 \colon x_i \geq 0 , a \leq \lVert x \rVert \leq b \}
  \,.
\]
For $p,q \in \R$ with $q \geq 0$, compute
\[
  A_i = \int_D x_i^q \, \lVert x \rVert^p \, \d x \,.
\]

\item Calculate the volume of the domain in $\R^3$ bounded by the
surfaces $z=x^2+y^2$, $y=x^2$, $y=1$, and $z=0$.


\end{enumerate}
\end{document}
