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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Tuesday, October 17, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 2}
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\firstpageheader{\bfseries \myCourseNumber \enspace \myAssignmentTitle \\ Due \myAssignmentDueDate }{}{}
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\begin{questions}

\question Change the order integration and evaluate:
\begin{displaymath}
	\int_{y=0}^1 \int_{x=y}^1 \sin\left(x^2\right) \, dx \, dy .
\end{displaymath}

\question Change the order integration and evaluate:
\begin{displaymath}
	\int_{y=0}^1 \int_{x=\sqrt{y}}^1 e^{x^3} \, dx \, dy .
\end{displaymath}

\question Let $D = [-1,1] \times [-1,2]$. \enspace Use the mean value inequality to show that
\begin{displaymath}
	1 \leq \iint_D \frac{1}{x^2 + y^2 + 1} \, dx \, dy \leq 6 . 
\end{displaymath}

\question Compute \enspace $\displaystyle \iint_D f(x,y) \, dA$, \enspace where \enspace $f(x,y) = y^2 \sqrt{x}$ \enspace and $D$ is the set of \enspace $(x,y)$ \enspace such that

$x > 0$, \enspace $y > x^2$, \enspace and \enspace $y < 10 - x^2$.

\question Perform the indicated integration over the given box:
\begin{displaymath}
	\iiint_B z \, e^{x+y} \, dx \, dy \, dz ; \quad B = [0,1] \times [0,1] \times [0,1] .
\end{displaymath}

\question Find the volume of the solid bounded by \enspace $x^2 + 2 y^2 = 2$, \enspace $z = 0$, \enspace and \enspace $x + y + 2 z = 2$.

\question Evaluate the integral \enspace $\displaystyle \iiint_W z \, dx \, dy \, dz$; \enspace where $W$ is the region bounded by \enspace $x = 0$, \enspace $y = 0$,

$z = 0$, \enspace $z = 1$, \enspace and the cylinder \enspace $x^2 + y^2 = 1$, \enspace with \enspace $x \geq 0$, \enspace $y \geq 0$.

\question Let \enspace $S^* = (0, 1] \times [0, 2 \pi)$ \enspace and define \enspace $T(r, \theta) = (r \cos(\theta), \; r \sin(\theta))$.
\begin{parts}
	\part Determine the image set $S = T(S^*)$.
	\part Show that $T$ is one-to-one on $S^*$.
\end{parts}

\question Let $D^*$ be the parallelogram with vertices at \enspace $(-1,3)$, \enspace $(0,0)$, \enspace $(2,-1)$, \enspace and \enspace $(1,2)$. \enspace Let $D$ \enspace be the rectangle \enspace $D = [0,1] \times [0,1]$. \enspace Find a $T$ such that $D$ is the image set of $D^*$ under $T$; that is, $D = T(D^*)$.

\question Let \enspace $T : \mathbb{R}^3 \to \mathbb{R}^3$ \enspace be the spherical coordinate mapping defined by \enspace $(\rho, \phi, \theta) \mapsto (x,y,z)$, \enspace where
\begin{displaymath}
x = \rho \sin(\phi) \cos(\theta), \qquad y = \rho \sin(\phi) \sin(\theta), \qquad z = \rho \cos(\phi) .
\end{displaymath}
Let $D^*$ be the set of points \enspace $(\rho, \phi, \theta)$ \enspace such that \enspace $\rho \in [0,1]$, \enspace $\phi \in [0, \pi]$, \enspace $\theta \in [0, 2 \pi]$.
\begin{parts}
\part Find \enspace $D = T(D^*)$.
\part Is $T$ one-to-one?  If not, can we eliminate a subset $S \subseteq D^*$ so that $T$ is one-to-one on the remainder \enspace $D^* \setminus S = \left\{ (x,y,z) \in D^* \mid (x,y,z) \not\in S \right\}$?
\end{parts}

\end{questions}

\end{document}
