@@ -70,7 +70,7 @@ \subsection{Integration Rules}
7070\noindent\lxAddClass {columns3}\parbox [t]{.23\linewidth }{%
7171\begin {enumerate }
7272\item $ \ds \int c\cdot f(x)\dd x=c\int f(x)\dd x$
73- \item $ \ds \int f(x)\pm g(x)\dd x=\\ [\defaultaddspace ]
73+ \item $ \ds \int \left ( f(x)\pm g(x) \right )\dd x=\\ [\defaultaddspace ]
7474\null\hfill\ds \int f(x)\dd x\pm\int g(x)\dd x$
7575\item $ \ds \int 0 \dd x=C$
7676\item $ \ds \int 1 \dd x=x+C$
@@ -98,7 +98,7 @@ \subsection{Integration Rules}
9898\end {enumerate }}\hfill
9999\parbox [t]{.36\linewidth }{%
100100\begin {enumerate }\setcounter {enumi}{22}
101- \item $ \ds \int \frac 1 {x\sqrt {x^2-a^2}}\dd x=\frac 1 a \ sec ^{-1}\left (\frac {\abs x}{a}\right )+C$
101+ \item $ \ds \int \frac 1 {x\sqrt {x^2-a^2}}\dd x=\frac 1 { \abs a} \ sec ^{-1}\left (\frac {\abs x}{\abs a}\right )+C$
102102\item $ \ds \int \cosh x\dd x=\sinh x+C$
103103\item $ \ds \int \sinh x\dd x=\cosh x+C$
104104\item $ \ds \int \tanh x\dd x=\ln (\cosh x)+C$
@@ -252,7 +252,7 @@ \subsubsection*{Sum to Product Formulas}
252252\end {align* }
253253\end {minipage }%
254254\begin {minipage }[t]{.3\linewidth }
255- \subsubsection* {Power-- Reducing Formulas }
255+ \subsubsection* {Power-Reducing Formulas }
256256\begin {align* }
257257\sin ^2 x &= \frac {1-\cos 2x}{2} & \vphantom {\left (\frac 11\right )}\\
258258\cos ^2 x &= \frac {1+\cos 2x}{2} & \vphantom {\left (\frac 11\right )}\\
@@ -293,12 +293,18 @@ \subsubsection*{Angle Sum/Difference Formulas}
293293
294294\clearpage
295295
296+ \mbox {}\vfill
297+
296298\subsection {Areas and Volumes }
297299
298- % this acts like there's an arraystretch, but I'm not sure why
299- \begin {tabular }{p{.22\linewidth }p{.22\linewidth }p{.22\linewidth }p{.22\linewidth }}
300- \paragraph {Triangles }
301- \vbox {\begin {flalign* }
300+ \vfill
301+
302+ \begin {tabular }{
303+ p{.22\linewidth }p{.2\linewidth } @{\hskip 4em}
304+ p{.22\linewidth }p{.2\linewidth }
305+ }
306+ \parbox [t]{\linewidth }{\begin {flalign* }
307+ &\textbf {Triangles }\\
302308 &h=a\sin\theta &\\
303309 &\text {Area} = \frac 12bh \\
304310 &\text {Law of Cosines:} \\
@@ -313,8 +319,8 @@ \subsection{Areas and Volumes}
313319 \draw (2,.2) -- (1.8,.2) -- (1.8,0);
314320 \end {tikzpicture }
315321 &
316- \paragraph { Right Circular Cone }
317- \vbox { \begin { flalign* }
322+ \parbox [t]{ \linewidth }{ \begin { flalign* }
323+ & \textbf { Right Circular Cone } \\
318324 &\text {Volume} = \frac 13 \pi r^2h &\\
319325 &\text {Surface Area} = \\
320326 &\pi r\sqrt {r^2+h^2} +\pi r^2
@@ -332,8 +338,8 @@ \subsection{Areas and Volumes}
332338 \draw [fill=black] (0,0) circle (1pt);
333339 \end {tikzpicture }
334340 \\
335- \paragraph { Parallelograms }
336- \vbox { \begin { flalign* }
341+ \parbox [t]{ \linewidth }{ \begin { flalign* }
342+ & \textbf { Parallelograms } \\
337343 &\text {Area} = bh &
338344 \end {flalign* }}
339345 &
@@ -344,8 +350,8 @@ \subsection{Areas and Volumes}
344350 \draw (.8,0) -- (.8,.2) -- (1,.2);
345351 \end {tikzpicture }
346352 &
347- \paragraph { Right Circular Cylinder }
348- \vbox { \begin { flalign* }
353+ \parbox [t]{ \linewidth }{ \begin { flalign* }
354+ & \textbf { Right Circular Cylinder } \\
349355 &\text {Volume} = \pi r^2h &\\
350356 &\text {Surface Area} = \\
351357 &2\pi rh +2\pi r^2
@@ -363,8 +369,8 @@ \subsection{Areas and Volumes}
363369 \draw [fill=black] (0,2.5) circle (1pt);
364370 \end {tikzpicture }
365371 \\
366- \paragraph { Trapezoids }
367- \vbox { \begin { flalign* }
372+ \parbox [t]{ \linewidth }{ \begin { flalign* }
373+ & \textbf { Trapezoids } \\
368374 & \text {Area} = \frac 12(a+b)h &
369375 \end {flalign* }}
370376 &
@@ -375,8 +381,8 @@ \subsection{Areas and Volumes}
375381 \draw (1.3,0) -- (1.3,.2) -- (1.5,.2);
376382 \end {tikzpicture }
377383 &
378- \paragraph { Sphere }
379- \vbox { \begin { flalign* }
384+ \parbox [t]{ \linewidth }{ \begin { flalign* }
385+ & \textbf { Sphere } \\
380386 &\text {Volume} = \frac 43\pi r^3 &\\
381387 &\text {Surface Area} = 4\pi r^2
382388 \end {flalign* }}
@@ -392,8 +398,8 @@ \subsection{Areas and Volumes}
392398 \draw [fill=black] (0,0) circle (1pt);
393399 \end {tikzpicture }
394400 \\
395- \paragraph { Circles }
396- \vbox { \begin { flalign* }
401+ \parbox [t]{ \linewidth }{ \begin { flalign* }
402+ & \textbf { Circles } \\
397403 &\text {Area} = \pi r^2 &\\
398404 &\text {Circumference} = 2\pi r
399405 \end {flalign* }}
@@ -405,8 +411,8 @@ \subsection{Areas and Volumes}
405411 \draw [fill=black] (0,0) circle (1pt);
406412 \end {tikzpicture }
407413 &
408- \paragraph { General Cone }
409- \vbox { \begin { flalign* }
414+ \parbox [t]{ \linewidth }{ \begin { flalign* }
415+ & \textbf { General Cone } \\
410416 &\text {Area of Base} = A &\\
411417 &\text {Volume} = \frac 13Ah
412418 \end {flalign* }}
@@ -427,8 +433,8 @@ \subsection{Areas and Volumes}
427433 \draw (1.5,-.75) node {$ A$ };
428434 \end {tikzpicture }
429435 \\
430- \paragraph { Sectors of Circles }
431- \vbox { \begin { flalign* }
436+ \parbox [t]{ \linewidth }{ \begin { flalign* }
437+ & \textbf { Sectors of Circles } \\
432438 &\theta \text { in radians} &\\
433439 &\text {Area} = \frac 12\theta r^2 \\
434440 &s=r\theta
@@ -443,8 +449,8 @@ \subsection{Areas and Volumes}
443449 \draw (0,0) node [shift={(15pt,8pt)}] {$ \theta $ };
444450 \end {tikzpicture }
445451 &
446- \paragraph { General Right Cylinder }
447- \vbox { \begin { flalign* }
452+ \parbox [t]{ \linewidth }{ \begin { flalign* }
453+ & \textbf { General Right Cylinder } \\
448454 &\text {Area of Base} = A &\\
449455 &\text {Volume} = Ah
450456 \end {flalign* }}
@@ -467,6 +473,8 @@ \subsection{Areas and Volumes}
467473 \end {tikzpicture }
468474\end {tabular }
469475
476+ \vfill\vfill\vfill
477+
470478\clearpage
471479
472480\subsection {Algebra }
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