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%  Spring semester, 2024, Wei-Cheng Wang.


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\begin{document}


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\noindent
{\scriptsize Calculus II, Spring 2024 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\noindent
{\scriptsize Thomas' Calculus Early Transcendentals 13ed} \hfill 

\begin{center}
\large
Study guide for quiz 11
\normalsize
\end{center}

\noindent

\vspace{-.3cm}
Quiz problems include both the lecture contents and homework problems.

\begin{enumerate}

\item
Section 15.3 
(Area by double integral) is a special case of Section 15.2
(Double integrals in general domain) and will not appear in quiz 11.

% 15.4: Double Integrals in Polar Form.
\item 
Section 15.4: 

Study why $dA = r dr d\theta$ in polar coordinates.
Practice how to determine the limits of integration
in $\displaystyle{ \int_{\theta_1}^{\theta_2} \int_{f_1(\theta)}^{f_2(\theta)} (\cdots) r dr d\theta }$ 
as in Examples 2-6 of section 15.4.
More specifically:
\begin{enumerate}
\item 
Given a domain $R$ in the $x-y$ plane,
practice drawing $\theta = C$ lines in $R$.
The end points of these lines are lower limit
(the near end point) and upper limit (the far end point)
of the integration $rdr$.
The end points for $d\theta$ are smallest and largest
$C$ among these $\theta = C$ lines.

\item
The end points of the lines $\theta = C$ (that is, the lower limit and upper limit of $rdr$)
must be expressed as $r = f_1(\theta)$ and $r = f_2(\theta)$.

Given a simple curve $F(x,y)=0$ (such as a line or a circle),
use the substitution $x=r\cos\theta$, $y=r\cos\theta$ to
express it as $r = f(\theta)$.
Examples: $x=1$, $y= -3$, $x+y = 1$, $x^2+y^2 = 4$, etc.
\end{enumerate}

% 15.5: Triple Integrals in Rectangular Coordinates.
\item
Section 15.5: 

Practice how to determine the limits of integration
for triple integrals in rectangular coordinates.
For example, which cross section (
$\{x={\rm constant} \}$, 
$\{y={\rm constant} \}$ or
$\{x={\rm constant} \}$) 
is needed for $dx dy dz$? which cross section is needed
for $dz dx dy$? etc.

On a cross section, the triple integral reduces to double integral
for the first two integration variables.

Note that, the upper and lower limits of the first (inner)
variable may depend on the second (middle) and third (outer) variables.
The upper and lower limits of the second
variable may depend on the third variable.

Practice this on corresponding examples and exercises
in section 15.5.

% 15.7: Triple Integrals in Cylindrical and Spherical Coordinates.
% \item
% Section 15.7: 

% Practice on drawing cross section 
% $\{r={\rm constant} \}$, 
% $\{\theta ={\rm constant} \}$ and
% $\{z={\rm constant} \}$ in cylindrical coordinates.
% Which one 
% is needed for $dr d\theta dz$? which one is needed
% for $dz dr d\theta$? etc.

% On a cross section, the triple integral reduces to double integral
% for the first two integration variables.
% For example, on a $\{ z={\rm constant} \}$ cross section, it reduces to
% double integral $dr d\theta$. One can then follow item 4 above to
% determine the upper and lower limits of integration. 
% 
% Repeat the same practice in spherical coordinates.
% 
% Also note that, the upper and lower limits of the first
% variable may depend on the second and third variables.
% The upper and lower limits of the second
% variable may depend on the third variable.

\end{enumerate}

\end{document}
