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%  Spring semester, 2024, Wei-Cheng Wang.


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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 12
\normalsize
\end{center}

\noindent

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

\begin{enumerate}

% 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.

% 15.8: Substitutions in Multiple Integrals.
\item
Section 15.8: 

Study the meaning of the Jacobian and
memorize the formula
both in double and triple integration.

% 15.8: Substitutions in Multiple Integrals.
\item
Section 15.8: 

Suppose that a change of variables between $(x,y,z)$
and $(u,v,w)$ is given, study how to change the
lower and upper limits of integration for $(x,y,z)$
into
lower and upper limits of integration for $(u,v,w)$.
See the examples in section 15.8.


\end{enumerate}

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