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\begin{document}

\input{213-Lecture-Macros}		%% 	macros, settings, et.c


\title{Math 213 - Triple Integrals in Spherical Coordinates}			%%	title page
\author{Peter A. Perry}
\institute{University of Kentucky}
\date{October 30, 2019}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	
%	Due Date Reminders
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Reminders}

\begin{itemize}

\item	Homework C3 on section 15.7 (triple integrals in cyindrical coordinates) is due tonight
\item	Quiz \#7 on sections 15.3 and 15.6 takes place tomorrow in recitation
\item	Homework C4 on section 15.8 (triple integrals in spherical coordinates) is due 
Friday
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Unit Outline
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Unit III: Multiple Integrals, Vector Fields}
\begin{itemize}

\item[{}]			Double Integrals in Polar Coordinates
\item[{}]			Triple Integrals (Part I)
\item[{}]			Triple Integrals (Part II)
\item[{}]			Triple Integrals in Cylindrical Coordinates
\begingroup
\color{red}
\item[{}]			Triple Integrals in Spherical Coordinates
\endgroup
\item[{}]			Change of Variables, Part I
\item[{}]			Change of Variables, Part II

\medskip

\item[{}]			Vector Fields
\item[{}]			Line Integrals (Scalar functions)
\item[{}]			Line Integrals (Vector functions)

\medskip

\item[{}]			Exam III Review
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Learning Goals
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Learning Goals}

\begin{frame}
\frametitle{Goals of the Day}
\begin{itemize}
\item	Know how to locate points and describe regions in spherical coordinates
\item	Know how to evaluate triple integrals in spherical coordinates
\end{itemize}
\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%		New material begins here...
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Spherical Coordinates}

\begin{frame}
\frametitle{Spherical Coordinates}
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[scale=0.6]
\drawxyz{4}
%\node 	at					(0.3,-0.2,0.3)	{\scriptsize{$O$}};
\draw[black,fill=black]	(4,3,5)	circle(0.05cm);
\node[above] at			(4,3,5)	{\scriptsize{$P$}};
\draw[black,fill=black]	(4,0,5)	circle(0.05cm);
\node[below] at			(4,0,5)	{\scriptsize{$P'$}};
\draw[dashed]			(0,0,0)	--	(4,0,5);
\draw[dashed]			(4,0,5)	--	(4,3,5);
\uncover<2->
{
\draw[dashed,cyan]	(0,0,0)	--	(4,3,5);
\node[below]	at		(3,2.25,3.75)		{\color{cyan}\scriptsize{$\rho$}};
}
\uncover<3->		
{
\draw[->,>=stealth,cyan]		(0,1.5,0)	arc(90:40:2);
\node at (0.5,0.75,0)					{\color{cyan}\scriptsize{$\phi$}};
}
\uncover<4->
{
\draw[->,>=stealth,cyan]		(0,0,2)	arc(250:290:2 and 1.5);
\node	at		(0.375,0,1)				{\color{cyan}\scriptsize{$\theta$}};
}
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{0.5\textwidth}
\footnotesize
The spherical coordinates $(\rho,\theta,\phi)$ of a point $P$  in three-dimensional space with projection $P'$ on the $xy$-plane are:
\medskip
\begin{itemize}
\uncover<2->
{
\item	$\rho = \sqrt{x^2+y^2+z^2}$, the distance $|\overrightarrow{OP}|$
}
\medskip
\uncover<3->
{
\item	$\phi$, the angle that the vector $\overrightarrow{OP}$ makes with the $z$-axis
}
\medskip
\uncover<4->
{
\item	$\theta$, the angle that the vector $\overrightarrow{OP'}$ makes with the $x$-axis
}
\end{itemize}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Cartesian to Spherical and Back Again}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[scale=0.6]
\drawxyz{4}
%\node 	at					(0.3,-0.2,0.3)	{\scriptsize{$O$}};
\draw[black,fill=black]	(4,3,5)	circle(0.05cm);
\node[above] at			(4,3,5)	{\scriptsize{$P$}};
\draw[black,fill=black]	(4,0,5)	circle(0.05cm);
\node[below] at			(4,0,5)	{\scriptsize{$P'$}};
\draw[dashed]	(0,0,0)	--	(4,3,5);
\node[below]	at		(3,2.25,3.75)		{\scriptsize{$\rho$}};
\draw[->,>=stealth]		(0,1.5,0)	arc(90:40:2);
\node at (0.5,0.75,0)					{\scriptsize{$\phi$}};
\only<1>
{
\draw[dashed]			(0,0,0)	--	(4,0,5);
\draw[dashed]			(4,0,5)	--	(4,3,5);
}
\draw[->,>=stealth]		(0,0,2)	arc(250:290:2 and 1.5);
\node	at		(0.375,0,1)				{\scriptsize{$\theta$}};
\uncover<2->
{
\draw[dashed,cyan]	(0,0,0)	--	(4,0,5);
\draw[dashed,cyan]	(4,0,5)	--	(4,3,5);
\node[right] at (4,1.5,5)				{\color{cyan}\scriptsize{$\rho \cos \phi$}};
\node[left] at (3,0,4)					{\color{cyan}\scriptsize{$\rho \sin \phi$}};
}
\end{tikzpicture}

\bigskip
\end{minipage}
\quad
\begin{minipage}{0.55\textwidth}
\footnotesize
Going over:
\begin{align*}
\rho 	&=	\sqrt{x^2+y^2+z^2}	\\
\tan \theta					&=	\frac{y}{x}\\
\cos \phi					&=	\frac{z}{\rho}
\end{align*}

\medskip

\uncover<2->
{
Coming back:
\begin{align*}
x	&=	\rho \sin \phi \cos \theta\\
y	&=	\rho \sin \phi \sin \theta\\
z	&=	\rho \cos \phi
\end{align*}
}

\uncover<3->
{
\begin{enumerate}
\item	Find the spherical coordinates of the point $(1,\sqrt{3},4)$
\item	Find the cartesian coordinates of the point $(4,\pi/4,\pi/2)$
\end{enumerate}
}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Regions in Spherical Coordinates}
\footnotesize
Match each of the following surfaces with its graph in $xyz$ space
\begin{enumerate}
\item	$\theta=c$
\item	$\rho = 5$
\item $\phi  = c, \quad 0< c < \pi/2$
\end{enumerate}

\bigskip

\begin{center}
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.3]
\draw[thick,->,>=stealth]	(0,0,0)	--	++(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	++(0,5,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	++(0,0,7);
%%
\node[below] at	(0,0,7)		{\scriptsize{$x$}};
\node[right] at		(5,0,0)		{\scriptsize{$y$}};
\node[above] at	(0,5,0)		{\scriptsize{$z$}};
\draw[cyan,opacity=0.75,shading=ball]	(0,0,-0.5)	circle(3cm);
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.3]
\drawxyz{5}
\draw[cyan,fill=cyan!50,opacity=0.5] (0,-1,0)	--	++(4,0,4)	
--	++(0,5,0)	--	++(-4,0,-4)	
--	++(0,-5,0);
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.3]
\draw[thick,->,>=stealth]	(0,0,0)	--	++(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	++(0,6,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	++(0,0,5);
%%
\node[below] at	(0,0,5)		{\scriptsize{$x$}};
\node[right] at		(5,0,0)		{\scriptsize{$y$}};
\node[above] at	(0,6,0)		{\scriptsize{$z$}};
\draw[cyan,fill=cyan!80,opacity=0.5]	(0,0,0)	--	++(-2,4,0)	--
	(-2,4,0) arc(180:360:2cm and 0.9cm)	--	++(-2,-4,0);
\draw[cyan,left color=cyan!70,right color=cyan!10,opacity=0.5]
	(-2,4,0)	arc(180:540:2cm and 0.9cm);
\end{tikzpicture}
\end{minipage}
\end{center}
\end{frame}

\begin{frame}
\frametitle{A Spherical Wedge}
\footnotesize
The region
$$ E = \{ (\rho, \theta, \phi): a \leq \rho \leq b, \, \alpha \leq \theta \leq \beta, \, c \leq \phi \leq d \} $$
is a \emph{spherical wedge}. What does it look like?

\vfill

\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[scale=0.55]
\draw[thick,->,>=stealth]	(0,0,0)	--	++(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	++(0,5,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	++(0,0,7);
%%
\node[below] at	(0,0,7)		{\scriptsize{$x$}};
\node[right] at		(5,0,0)		{\scriptsize{$y$}};
\node[above] at	(0,5,0)		{\scriptsize{$z$}};
\only<1>
{
\draw[white,fill=orange!70,opacity=0.5]		(0,0,-0.2)	circle(2cm);
\draw[white,fill=orange!30,opacity=0.2]		(0,0,-0.35)	circle(3.5cm);
}
\only<2>
{
%% segment between spheres of radius 2 and 3.5 betwen theta=45and theta=90
%% side at theta = 45
\draw[orange,fill=orange!50]
			plot[domain=180:0,variable=\phi]	
					(	
						{2*sin(\phi)*sin(45)},	
						{2*cos(\phi)},	
						{2*sin(\phi)*cos(45)}
					)	
		--	(0,3.5,0)	
		--	plot[domain=0:180,variable=\phi]	
					(
						{3.5*sin(\phi)*sin(45)},	
						{3.5*cos(\phi)}, 	
						{3.5*sin(\phi)*cos(45)}
					)	
		-- (0,-2,0);
%% segment of sphere of radius 3.5 between theta=45 and theta=90
\draw[orange,fill=orange!20]	
				plot[domain=180:0,variable=\phi]
					(
						{3.5*sin(\phi)*sin(90)},	
						{3.5*cos(\phi)}, 	
						{3.5*sin(\phi)*cos(90)}
					)
			--	plot[domain=0:180,variable=\phi]	
					(
						{3.5*sin(\phi)*sin(45)},	
						{3.5*cos(\phi)}, 	
						{3.5*sin(\phi)*cos(45)}
					);	
%% add dotted line at equator
\draw[orange,dashed]
			plot[domain=45:90,variable=\theta]
					(
						{3.5*sin(\theta)},
						{0},
						{3.5*cos(\theta)}
					);
}
\only<3>
{
%% wedge from phi=70 to phi=15 with theta=45:90
%% segment between spheres of radius 2 and 3.5 and theta=45
\draw[orange,fill=orange!50]
			plot[domain=70:15,variable=\phi]
				(
					{2*sin(\phi)*sin(45)},
					{2*cos(\phi)},
					{2*sin(\phi)*cos(45)}
				)
		--	plot[domain=2:3.5,variable=\rho]
				(
					{\rho*sin(15)*sin(45)},
					{\rho*cos(15)},
					{\rho*sin(15)*sin(45)}
				)
		--	plot[domain=15:70,variable=\phi]
				(
					{3.5*sin(\phi)*sin(45)},
					{3.5*cos(\phi)},
					{3.5*sin(\phi)*sin(45)}
				)
		--	plot[domain=3.5:2,variable=\rho]
				(
					{\rho*sin(70)*sin(45)},
					{\rho*cos(70)},
					{\rho*sin(70)*cos(45)}
				);
%% front part of segment at rho = 3.5
\draw[orange,fill=orange!20]
			plot[domain=45:90,variable=\theta]
				(
					{3.5*sin(70)*sin(\theta)},
					{3.5*cos(70)},
					{3.5*sin(70)*cos(\theta)}
				)
		--	plot[domain=70:15,variable=\phi]	
				(
					{3.5*sin(\phi)},
					{3.5*cos(\phi)},
					{0}
				)
		--	plot[domain=90:45,variable=\theta]
				(
					{3.5*sin(15)*sin(\theta)},
					{3.5*cos(15)},
					{3.5*sin(15)*cos(\theta)}
				)
		--	plot[domain=15:70,variable=\phi]
				(
					{3.5*sin(\phi)*sin(45)},
					{3.5*cos(\phi)},
					{3.5*sin(\phi)*cos(45)}
				);
}
\end{tikzpicture}
				
					
\end{minipage}
\quad
\begin{minipage}{0.5\textwidth}
\begin{itemize}
\uncover<1->
{
\item	$a \leq \rho \leq b$ means the shape lies between spheres of radius $a$ and $b$
}
\uncover<2->
{
\item	$\alpha \leq \theta \leq \beta$ restricts the shape to a wedge-shaped region over the $xy$ plane
}
\uncover<3->
{
\item	$c \leq \phi \leq d$ restricts the shape to the space between two cones about the $z$-axis
}
\end{itemize}
\end{minipage}

\end{frame}

\begin{frame}
\frametitle{Describing Regions in Spherical Coordinates}
\footnotesize
Can you sketch each of these regions?

\medskip
\begin{enumerate}
\item	$0 \leq \rho \leq 1$, \quad $0 \leq \phi \leq \pi/6$, \quad $0 \leq \theta \leq \pi$
\item	$1 \leq \rho \leq 2$, \quad $\pi/2 \leq \phi \leq \pi$
\item	$2 \leq \rho \leq 4$, \quad $0 \leq \phi \leq \pi/3$, \quad $0 \leq \theta \leq \pi$
\end{enumerate}


\end{frame}


\section{Triple Integrals in Spherical Coordinates}
\begin{frame}
\frametitle{Triple Integrals in Spherical Coordinates}
\footnotesize
We need to find the volume of a small spherical wedge

\bigskip


\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}
%% to draw this picture, change phi range to 40:70
%% outer surface
\draw[orange,fill=orange!20]
			plot[domain=45:90,variable=\theta]
				(
					{3.5*sin(70)*sin(\theta)},
					{3.5*cos(70)},
					{3.5*sin(70)*cos(\theta)}
				)
		--	plot[domain=70:40,variable=\phi]	
				(
					{3.5*sin(\phi)},
					{3.5*cos(\phi)},
					{0}
				)
		--	plot[domain=90:45,variable=\theta]
				(
					{3.5*sin(40)*sin(\theta)},
					{3.5*cos(40)},
					{3.5*sin(40)*cos(\theta)}
				)
		--	plot[domain=40:70,variable=\phi]
				(
					{3.5*sin(\phi)*sin(45)},
					{3.5*cos(\phi)},
					{3.5*sin(\phi)*cos(45)}
				);
%% side toward xz plane
\draw[orange,fill=orange!50]
			plot[domain=70:40,variable=\phi]
				(
					{2*sin(\phi)*sin(45)},
					{2*cos(\phi)},
					{2*sin(\phi)*cos(45)}
				)
		--	plot[domain=2:3.5,variable=\rho]
				(
					{\rho*sin(40)*sin(45)},
					{\rho*cos(40)},
					{\rho*sin(40)*sin(45)}
				)
		--	plot[domain=40:70,variable=\phi]
				(
					{3.5*sin(\phi)*sin(45)},
					{3.5*cos(\phi)},
					{3.5*sin(\phi)*sin(45)}
				)
		--	plot[domain=3.5:2,variable=\rho]
				(
					{\rho*sin(70)*sin(45)},
					{\rho*cos(70)},
					{\rho*sin(70)*cos(45)}
				);
\uncover<2->
{
	\node[below]	at		(	
									{2.25*sin(70)*sin(45)},
									{2.25*cos(70)},
									{2.25*sin(70)*cos(45)}
								)
					{\scriptsize{\color{cyan}$d\rho$}};
}
\only<2>
{
	\draw[cyan,very thick]
			plot[domain=2:3.5,variable=\rho]
								(
									{\rho*sin(70)*sin(45)},
									{\rho*cos(70)},
									{\rho*sin(70)*cos(45)}
								);
}
\uncover<3->
{
	\node [right]	at		(
									{3.5*sin(55)},
									{3.5*cos(55)},
									{0}
								)
					{\scriptsize{\color{cyan}$\rho \, d\phi$}};			
}
\only<3>
{
		\draw[cyan,very thick]
				plot[domain=70:40,variable=\phi]
								(
									{3.5*sin(\phi)},
									{3.5*cos(\phi)},
									0
								);
}
\uncover<4->
{
	\node [right]	at		(
									{3.5*sin(70)*sin(67.5)},
									{3.5*cos(70)},
									{3.5*sin(70)*cos(67.5)}
								)
					{\scriptsize{\color{cyan}$\rho \sin \phi \, d\theta$}};
}
\only<4>
{
		\draw[cyan,very thick]
					plot[domain=45:90,variable=\theta]
								(
									{3.5*sin(70)*sin(\theta)},
									{3.5*cos(70)},
									{3.5*sin(70)*cos(\theta)}
								);
}
\end{tikzpicture}

\end{minipage}
\quad
\begin{minipage}{0.5\textwidth}
Volume comes from 
\begin{itemize}
\uncover<2->
{
\item	Change in $\rho$
}
\uncover<3->
{
\item	Change in $\phi$
}
\uncover<4->
{
\item	Change in $\theta$
}

\bigskip

$$ dV = \only<4>{\rho^2 \sin \phi} \only<3>{\rho} \uncover<2->{ \, d\rho} \uncover<3->{\, d\phi}\uncover<4->{\, d\theta} $$
\end{itemize}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Triple Integrals in Spherical Coordinates}

\footnotesize
\begin{multline*}
 \iint_E f(x,y,z) \, dV = \\
 \int_c^d \int_\alpha^\beta \int_a^b f(\rho \sin \phi \cos \theta,\rho \sin \phi \sin \theta, \rho cos \phi) 			\, 	{\color{red} \rho^2 \sin \phi} \,  d\rho \, d\theta \, d\phi
\end{multline*}
if $E$ is a spherical wedge
$$ E = \{ (\rho,\theta,\phi): a \leq \rho \leq b, \, \alpha \leq \theta \leq \beta, \, c \leq \phi \leq d \} $$

\vfill

\begin{enumerate}
\item	Find $\iiint_E y^2 z^2 \, dV$ if $E$ is the region above the cone $\phi=\pi/3$ and below the sphere $\rho=1$

\medskip


\item	Find $\iiint_E y^2 dV$ if $E$ is the solid hemisphere $x^2+y^2+z^2 \leq 9$, $y \geq 0$

\medskip

\item	Find $\iiint_E \sqrt{x^2+y^2+z^2} \, dV$ if $E$ lies above the cone $z=\sqrt{x^2+y^2}$ and between the spheres $\rho=1$ and $\rho=2$

\end{enumerate}
\end{frame}



\end{document}
