\documentclass[addpoints, 11pt]{exam}
\setlength{\headsep}{0.125in}
\setlength{\unitlength}{1in}
%
\pagestyle{head}
%
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
%
\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{Wednesday, October 24, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 3}
%
%
\begin{document}
%
\firstpageheader{\bfseries \myCourseNumber \enspace \myAssignmentTitle \\ Due \myAssignmentDueDate }{}{}
%
\runningheader{{\bfseries (page \textit{\thepage}\ of \textit{\numpages})}}{}{}
%
%\firstpagefooter{}{}{(This exam is worth {\numpoints} points.)}
%

\begin{questions}
    
\question Let $D$ be the unit disk: $x^2 + y^2 \leq 1$.  Evaluate $\displaystyle \iint_D \exp\left(x^2 + y^2\right) \, dx \, dy$.

\question Evaluate $\displaystyle \iint_D x^2 \, dx \, dy$ where $D$ is determined by the two conditions $0 \leq x \leq y$ and $x^2 + y^2 \leq 1$.

\question Evaluate $\displaystyle \iiint_W \sqrt{x^2 + y^2 + z^2} \, e^{-(x^2 + y^2 + z^2)} \, dx \, dy \, dz$, where $W$ is the solid bounded by the two spheres

$x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ with $0 < a < b$.

\question Evaluate $\displaystyle \iint_R (x + y) \, dx \, dy$, where $R$ is the rectangle in the $xy$-plane with vertices at $(0,1), (1,0), (3,4), (4,3)$.

\question Show that the path $\mathbf{c}(t) = \left(\sin(t), \cos(t), e^t\right)$ is a flow line of the vector field \newline $\mathbf{F}(x,y,z) = (y, -x, z)$.

\question Let $\mathbf{F}(x,y,z) = (y z, x z, x y)$.  Find a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that $\mathbf{F} = \nabla f$.

\question Evaluate the path integral $\displaystyle \int_{\mathbf{c}} f(x,y,z) \, ds$ with $f(x,y,z) = x+y+z$ and $\mathbf{c}(t) = (\sin(t), \cos(t), t)$ for $t \in \left[0, 2\pi\right]$.

\question Find the average $y$ coordinate of the points on the semicircle parametrized by $\mathbf{c} : [0,\pi] \to \mathbb{R}^3$ given by $\mathbf{c}(t) = (0, a \sin(t), a \cos(t))$ with $a > 0$.

\question Evaluate $\displaystyle \int_{\mathbf{c}} f \, ds$, where $f(x,y,z) = z$ and $\mathbf{c}(t) = (t \cos(t), t \sin(t), t)$ for $0 \leq t \leq t_0$.

\question Find the average $z$ coordinate on the path $\mathbf{c}(t) = (t \cos(t), t \sin(t), t)$ for $0 \leq t \leq t_0$.

\end{questions}

\end{document}
