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\author{CROSSANT (CROSSANT)}
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  /Title (calculus-ii.pdf)
  /Creator (Cheatography)
  /Author (CROSSANT (CROSSANT))
  /Subject (Calculus II Cheat Sheet)
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Calculus II Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{CROSSANT (CROSSANT)} via \textcolor{DarkBackground}{\uline{cheatography.com/186482/cs/38975/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}CROSSANT (CROSSANT) \\
  \uline{cheatography.com/crossant} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 28th May, 2023.\\
   Updated 11th July, 2024.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{4}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Series}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{{\bf{Series Type}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Infinite Series} \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Harmonic Series} \tn 
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\mymulticolumn{1}{x{3.833cm}}{Geometric Series} \tn 
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\mymulticolumn{1}{x{3.833cm}}{P-Series} \tn 
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\mymulticolumn{1}{x{3.833cm}}{Alternating Series} \tn 
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\mymulticolumn{1}{x{3.833cm}}{Telescoping Series} \tn 
% Row Count 7 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Alternating Series Estimation Theorem: If S\textasciitilde{}n\textasciitilde{}=Σ\textasciicircum{}n\textasciicircum{}\textasciitilde{}i=1\textasciitilde{}(-1)\textasciicircum{}n\textasciicircum{}b\textasciitilde{}n\textasciitilde{} or Σ\textasciicircum{}n\textasciicircum{}\textasciitilde{}i=1\textasciitilde{}(-1)\textasciicircum{}n-1\textasciicircum{}b\textasciitilde{}n\textasciitilde{} is the sum of an alternating series that converges, then |R\textasciitilde{}n\textasciitilde{}|=|S-S\textasciitilde{}n\textasciitilde{}|≤b\textasciitilde{}n+1\textasciitilde{} \newline Trigonometric functions like cos(nπ) or sin(nπ+π/2) act as sign alternators, like (-1)\textasciicircum{}n\textasciicircum{} \newline The Alternating Series Test does not show divergence, however, implementing the test requires a Test For Divergence, which does show divergence}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Series Tests}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{{\bf{Test Type}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Test for Divergence} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Integral Test} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{(Direct) Comparison Test} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Limit Comparison Test} \tn 
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% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Ratio Test} \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Root Test} \tn 
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\mymulticolumn{1}{x{3.833cm}}{Absolute/Conditional Convergence} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{For the series listed, assume each series to be an infinite series starting at n=k: Σ\textasciicircum{}∞\textasciicircum{}\textasciitilde{}n=k\textasciitilde{}=Σ \newline If Test for Divergence passes (lim \textasciitilde{}n-\textgreater{}∞\textasciitilde{} =0), use another test \newline The symbol {[} ⟺ {]} represents the relationship "if and only if" (often abbreviated to "iff"), meaning both sides of the statement must be true at the same time, or false at the same time \newline If a test is inconclusive, use another test}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.68458 cm} p{0.65825 cm} p{0.65825 cm} p{0.63192 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Special Series}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Series}} & {\bf{Summation Form}} & {\bf{First five terms}} & {\bf{Radius and Interval of Convergence}} \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
Power Series centered at a & ΣC\textasciitilde{}n\textasciitilde{}(x-a)\textasciicircum{}n\textasciicircum{} & C\textasciitilde{}0\textasciitilde{}+C\textasciitilde{}1\textasciitilde{}(x-a)+C\textasciitilde{}2\textasciitilde{}(x-a)\textasciicircum{}2\textasciicircum{}+C\textasciitilde{}3\textasciitilde{}(x-a)\textasciicircum{}3\textasciicircum{}+C\textasciitilde{}4\textasciitilde{}(x-a)\textasciicircum{}4\textasciicircum{}+... &  \tn 
% Row Count 11 (+ 6)
% Row 2
\SetRowColor{LightBackground}
Taylor Series centered at a & Σf\textasciicircum{}(n)\textasciicircum{}(a)(x-a)\textasciicircum{}n\textasciicircum{}/n! & f(a)+f'(a)(x-a)+f''(a)(x-a)\textasciicircum{}2\textasciicircum{}/2!+f'''(a)(x-a)\textasciicircum{}3\textasciicircum{}/3!+f\textasciicircum{}(4)\textasciicircum{}(a)(x-a)\textasciicircum{}4\textasciicircum{}/4!+... & R\textgreater{}|x-a| \tn 
% Row Count 19 (+ 8)
% Row 3
\SetRowColor{white}
Maclaurin Series (Taylor Series centered at 0) & Σf\textasciicircum{}(n)\textasciicircum{}(0)(x-0)\textasciicircum{}n\textasciicircum{}/n!=Σf\textasciicircum{}(n)\textasciicircum{}(0)x\textasciicircum{}n\textasciicircum{}/n! & f(0)+f'(0)x+f''(0)x\textasciicircum{}2\textasciicircum{}/2!+f'''(0)x\textasciicircum{}3\textasciicircum{}/3!+f\textasciicircum{}(4)\textasciicircum{}(0)x\textasciicircum{}4\textasciicircum{}/4!+... &  \tn 
% Row Count 26 (+ 7)
% Row 4
\SetRowColor{LightBackground}
1/(1-x) & Σx\textasciicircum{}n\textasciicircum{} & 1+x+x\textasciicircum{}2\textasciicircum{}+x\textasciicircum{}3\textasciicircum{}+x\textasciicircum{}4\textasciicircum{}+... & R=1, I=(-1,1) \tn 
% Row Count 29 (+ 3)
% Row 5
\SetRowColor{white}
e\textasciicircum{}x\textasciicircum{} & Σx\textasciicircum{}n\textasciicircum{}/n! & 1+x+x\textasciicircum{}2\textasciicircum{}/2!+x\textasciicircum{}3\textasciicircum{}/3!+x\textasciicircum{}4\textasciicircum{}/4!+... & R=∞, I=(-∞,∞) \tn 
% Row Count 33 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{3.833cm}{p{0.68458 cm} p{0.65825 cm} p{0.65825 cm} p{0.63192 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Special Series (cont)}}  \tn
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\SetRowColor{LightBackground}
ln(1+x) & Σ(-1)\textasciicircum{}n+1\textasciicircum{}x\textasciicircum{}n\textasciicircum{}/n & x-x\textasciicircum{}2\textasciicircum{}/2+x\textasciicircum{}3\textasciicircum{}/3-x\textasciicircum{}4\textasciicircum{}/4+x\textasciicircum{}5\textasciicircum{}/5+... & R=1, I=(-1,1{]} \tn 
% Row Count 4 (+ 4)
% Row 7
\SetRowColor{white}
arctan(x) & Σ(-1)\textasciicircum{}n\textasciicircum{}x\textasciicircum{}2n+1\textasciicircum{}/(2n+1) & x-x\textasciicircum{}3\textasciicircum{}/3+x\textasciicircum{}5\textasciicircum{}/5-x\textasciicircum{}7\textasciicircum{}/7+x\textasciicircum{}9\textasciicircum{}/9+... & R=1, I={[}-1,1{]} \tn 
% Row Count 8 (+ 4)
% Row 8
\SetRowColor{LightBackground}
sin(x) & Σ(-1)\textasciicircum{}n\textasciicircum{}x\textasciicircum{}2n+1\textasciicircum{}/(2n+1)! & x-x\textasciicircum{}3\textasciicircum{}/3!+x\textasciicircum{}5\textasciicircum{}/5!-x\textasciicircum{}7\textasciicircum{}/7!+x\textasciicircum{}9\textasciicircum{}/9!+... & R=∞, I=(-∞,∞) \tn 
% Row Count 12 (+ 4)
% Row 9
\SetRowColor{white}
cos(x) & Σ(-1)\textasciicircum{}n\textasciicircum{}x\textasciicircum{}2n\textasciicircum{}/(2n)! & 1-x\textasciicircum{}2\textasciicircum{}/2!+x\textasciicircum{}4\textasciicircum{}/4!-x\textasciicircum{}6\textasciicircum{}/6!+x\textasciicircum{}8\textasciicircum{}/8!+... & R=∞, I=(-∞,∞) \tn 
% Row Count 16 (+ 4)
% Row 10
\SetRowColor{LightBackground}
(1+x)\textasciicircum{}k\textasciicircum{} & Σ(\textasciicircum{}k\textasciicircum{}\textasciitilde{}n\textasciitilde{})x\textasciicircum{}n\textasciicircum{}=Σ((k(k-1)(k-2)(k-3)...(k-n+1))/n!)x\textasciicircum{}n\textasciicircum{} & 1+kx+k(k-1)x\textasciicircum{}2\textasciicircum{}/2!+k(k-1)(k-2)x\textasciicircum{}3\textasciicircum{}/3!+k(k-1)(k-2)(k-3)x\textasciicircum{}4\textasciicircum{}/4!+... & R=1 \tn 
% Row Count 23 (+ 7)
% Row 11
\SetRowColor{white}
Taylor's \seqsplit{Inequality} & |R\textasciitilde{}n\textasciitilde{}(x)|≤M|x-a|\textasciicircum{}n+1\textasciicircum{}/(n+1)!, given M≥|f\textasciicircum{}(n+1)\textasciicircum{}(x)| for all |x-a|≤d &  &  \tn 
% Row Count 31 (+ 8)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\SetRowColor{LightBackground}
\mymulticolumn{4}{x{3.833cm}}{For the series listed, assume each series to be an infinite series starting at n=0: Σ\textasciicircum{}∞\textasciicircum{}\textasciitilde{}n=0\textasciitilde{}=Σ \newline Note that the formula for a Degree 1 Taylor Polynomial, T\textasciitilde{}1\textasciitilde{}(x), has the same formula as the Linear Approximation formula learned in Calculus I \newline f\textasciicircum{}(n)\textasciicircum{} means "the nth derivative of the function f" \newline n!=n(n-1)!=n(n-1)(n-2)!=n(n-1)(n-2)(n-3)!=... \newline n! = \seqsplit{n(n-1)(n-2)(n-3)...*3*2*1} \newline 0!=1, 1!=1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.40753 cm} x{2.02547 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Areas of Functions}}  \tn
% Row 0
\SetRowColor{LightBackground}
Between two functions & ∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} ((top function)-(bottom function))dA \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Enclosed by a polar function & $\frac{1}{2}$∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} f(θ)\textasciicircum{}2\textasciicircum{}dθ \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Between two polar functions & $\frac{1}{2}$∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} ((outer polar function)\textasciicircum{}2\textasciicircum{}-(inner polar function)\textasciicircum{}2\textasciicircum{})dθ \tn 
% Row Count 7 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Area enclosed by a polar function is with respect to the pole, which is the origin \newline  \newline Average value of a function: f\textasciitilde{}avg\textasciitilde{}=1/(b-a)∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} f(x)dx}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.0299 cm} x{2.4031 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Volumes of Solids of Revolution}}  \tn
% Row 0
\SetRowColor{LightBackground}
Disk & π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} (radius)\textasciicircum{}2\textasciicircum{}dV \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Washer & π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} (outer radius)\textasciicircum{}2\textasciicircum{} - (inner radius)\textasciicircum{}2\textasciicircum{}dV \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Cylindrical Shell & 2π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} (radius)(height)dV \tn 
% Row Count 5 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{For Cylindrical Shells: radius=x or y, and height=f(x) or g(y)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.09856 cm} x{2.33444 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Arc Lengths}}  \tn
% Row 0
\SetRowColor{LightBackground}
Function & ∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} {\bf{√}}(1+(f'(x))\textasciicircum{}2\textasciicircum{})dx \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Parametric Function & ∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} {\bf{√}}((x'(t))\textasciicircum{}2\textasciicircum{}+(y'(t))\textasciicircum{}2\textasciicircum{})dt \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Polar Function & ∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} {\bf{√}}(r(θ)\textasciicircum{}2\textasciicircum{}+(r'(θ))\textasciicircum{}2\textasciicircum{})dθ \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{For standard functions: f'(x)=dy/dx \newline For parametric functions: x'(t)=dx/dt and y'(t)=dy/dt \newline For polar functions: r'(θ)=dr/dθ}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.7165 cm} x{1.7165 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Surface Areas}}  \tn
% Row 0
\SetRowColor{LightBackground}
Function revolved about an axis & 2π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} (radius)(Arc Length component)ds \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Function revolved about y-axis & 2π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} x{\bf{√}}(1+(f'(x))\textasciicircum{}2\textasciicircum{})dx \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Function revolved about x-axis & 2π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} y{\bf{√}}(1+(g'(y))\textasciicircum{}2\textasciicircum{})dy \tn 
% Row Count 7 (+ 2)
% Row 3
\SetRowColor{white}
Parametric function of t revolved about y-axis & 2π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} f(x){\bf{√}}((x'(t))\textasciicircum{}2\textasciicircum{}+(y'(t))\textasciicircum{}2\textasciicircum{})dt \tn 
% Row Count 10 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Parametric function of t revolved about x-axis & 2π∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} g(y){\bf{√}}((x'(t))\textasciicircum{}2\textasciicircum{}+(y'(t))\textasciicircum{}2\textasciicircum{})dt \tn 
% Row Count 13 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{f'(x)=dy/dx, g'(y)=dx/dy, x'(t)=dx/dt, and y'(t)=dy/dt}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.13289 cm} x{2.30011 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Integration by Parts}}  \tn
% Row 0
\SetRowColor{LightBackground}
Indefinite Integral & ∫udv=uv-∫vdu \tn 
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Definite Integral & ∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} udv=uv|\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} -∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}b\textasciicircum{} vdu \tn 
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\mymulticolumn{2}{x{3.833cm}}{Integration by Parts is used to integrate integrals that have components multiplied together in their simplest form, often referred to as a "product rule for integrals" \newline  \newline Choosing the "dv" term depends on what will simplify the integral the best, while being relatively simple to integrate \newline  \newline The constant of integration does not need to be inserted until the integral has been fully simplified}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Trigonometric Integrals}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{\seqsplit{https://cheatography.com/crossant/cheat-sheets/integral-trigonometry/}% Row Count 2 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
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\begin{tabularx}{3.833cm}{x{1.75083 cm} x{1.68217 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Integration by Partial Fractions}}  \tn
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(px+q)/((x-a)(x-b)) & A/(x-a) + B/(x-b) \tn 
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(px+q)/(x-a)\textasciicircum{}2\textasciicircum{} & A/(x-a) + B/(x-a)\textasciicircum{}2\textasciicircum{} \tn 
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(px$^{\textrm{2}}$+qx+r)/((x-a)(x-b)(x-c)) & A/(x-a) + B/(x-b) + C/(x-c) \tn 
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\SetRowColor{white}
(px\textasciicircum{}2\textasciicircum{}+qx+r)/((x-a)\textasciicircum{}2\textasciicircum{}(x-b)) & A/(x-a) + B/(x-a)\textasciicircum{}2\textasciicircum{} + C/(x-b) \tn 
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(px\textasciicircum{}2\textasciicircum{}+qx+r)/((x-a)(x\textasciicircum{}2\textasciicircum{}+bx+c)) & A/(x-a) + Bx+C/(x\textasciicircum{}2\textasciicircum{}+bx+c) \tn 
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∫1/(a\textasciicircum{}2\textasciicircum{}+x\textasciicircum{}2\textasciicircum{}) dx & (1/a)arctan(x/a)+C \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Integration by Partial Fractions is used to simplify integrals of polynomial rational expressions into simpler fractions with a factored, irreducible denominator \newline  \newline The degree (highest power) of the numerator's polynomial must be less than the degree of the denominator's polynomial, otherwise, polynomial long division must be used before converting the expression into partial fractions}  \tn 
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\end{tabularx}
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\begin{tabularx}{3.833cm}{x{1.30454 cm} x{2.12846 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Improper Integrals}}  \tn
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∫\textasciicircum{}∞\textasciicircum{}\textasciitilde{}a\textasciitilde{} f(x)dx & lim \textasciitilde{}t-\textgreater{}∞\textasciitilde{} ∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}t\textasciicircum{} f(x)dx \tn 
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∫\textasciicircum{}b\textasciicircum{}\textasciitilde{}-∞\textasciitilde{} f(x)dx & lim \textasciitilde{}t-\textgreater{}-∞\textasciitilde{} ∫\textasciitilde{}t\textasciitilde{}\textasciicircum{}b\textasciicircum{} f(x)dx \tn 
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∫\textasciicircum{}∞\textasciicircum{}\textasciitilde{}-∞\textasciitilde{} f(x)dx & lim \textasciitilde{}t-\textgreater{}-∞\textasciitilde{} ∫\textasciitilde{}t\textasciitilde{}\textasciicircum{}c\textasciicircum{} f(x)dx + lim \textasciitilde{}t-\textgreater{}∞\textasciitilde{} ∫\textasciitilde{}x\textasciitilde{}\textasciicircum{}t\textasciicircum{} f(x)dx \tn 
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Convergence of ∫f(x)dx & lim \textasciitilde{}t-\textgreater{}±∞\textasciitilde{} =L \tn 
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Divergence of ∫f(x)dx & lim \textasciitilde{}t-\textgreater{}±∞\textasciitilde{} =±∞ or DNE \tn 
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\mymulticolumn{2}{x{3.833cm}}{Improper Integrals are integrals with bounds at infinity (Type 1) or discontinuous bounds (Type 2)}  \tn 
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\end{tabularx}
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Conic Sections}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{\seqsplit{https://cheatography.com/crossant/cheat-sheets/conic-sections/}% Row Count 2 (+ 2)
} \tn 
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\end{tabularx}
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\begin{tabularx}{3.833cm}{x{1.7165 cm} x{1.7165 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Parametric Curves and Polar Functions}}  \tn
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Parametric Curve C as a function of Parameter t & (x,y)=(f(t),g(t)) for t on {[}a,b{]} \tn 
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Slope at a given point & \seqsplit{dy/dx=(dy/dt)/(dx/dt)} \tn 
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Second derivative & d\textasciicircum{}2\textasciicircum{}y/dx\textasciicircum{}2\textasciicircum{}=(dy/dt)/(dx/dt)\textasciicircum{}2\textasciicircum{} \tn 
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Polar Curve C as a function of Parameter θ & (r,θ)=(r,θ±2πn)=(-r,θ±πn) \tn 
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Slope at a given point & \seqsplit{dy/dx=(dy/dθ)/(dx/dθ)} \tn 
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\seqsplit{Cartesian/Rectangular} to Polar coordinates & x=rcos(θ), y=rsin(θ) \tn 
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Polar to \seqsplit{Cartesian/Rectangular} coordinates & r\textasciicircum{}2\textasciicircum{}=x\textasciicircum{}2\textasciicircum{}+y\textasciicircum{}2\textasciicircum{} or r={\bf{√}}(x\textasciicircum{}2\textasciicircum{}+y\textasciicircum{}2\textasciicircum{}), tanθ=y/x or θ=arctan(y/x) \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{(dx/dt)≠0, (dx/dθ)≠0}  \tn 
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\end{tabularx}
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\begin{tabularx}{3.833cm}{x{1.40753 cm} x{2.02547 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Integral Approximations and Error Bounds}}  \tn
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Midpoint Rule & Δx(f(x̄\textasciitilde{}1\textasciitilde{})+f(x̄\textasciitilde{}2\textasciitilde{})+f(x̄\textasciitilde{}3\textasciitilde{})+...+f(x̄\textasciitilde{}n-1\textasciitilde{})+f(x̄\textasciitilde{}n\textasciitilde{})) \tn 
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Trapezoidal Rule & (Δx/2)(f(x\textasciitilde{}1\textasciitilde{})+2f(x\textasciitilde{}2\textasciitilde{})+2f(x\textasciitilde{}3\textasciitilde{})+...+2f(x\textasciitilde{}n-1\textasciitilde{})+f(x\textasciitilde{}1\textasciitilde{})) \tn 
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Simpson's Rule & (Δx/3)(f(x\textasciitilde{}1\textasciitilde{})+4f(x\textasciitilde{}2\textasciitilde{})+2f(x\textasciitilde{}3\textasciitilde{})+4f(x\textasciitilde{}4\textasciitilde{})+2f(x\textasciitilde{}5\textasciitilde{}))+...+2f(x\textasciitilde{}n-2\textasciitilde{})+4f(x\textasciitilde{}n-1\textasciitilde{})+f(x\textasciitilde{}n\textasciitilde{})) \tn 
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\SetRowColor{white}
Midpoint Rule Error Bound & |E\textasciitilde{}m\textasciitilde{}|≤k(b-a)\textasciicircum{}3\textasciicircum{}/24n\textasciicircum{}2\textasciicircum{}, k=f''(x)\textasciitilde{}max\textasciitilde{} on {[}a,b{]} \tn 
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\SetRowColor{LightBackground}
Trapezoidal Rule Error Bound & |E\textasciitilde{}t\textasciitilde{}|≤k(b-a)\textasciicircum{}3\textasciicircum{}/12n\textasciicircum{}2\textasciicircum{}, k=f''(x)\textasciitilde{}max\textasciitilde{} on {[}a,b{]} \tn 
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\SetRowColor{white}
Simpson's Rule Error Bound & |E\textasciitilde{}s\textasciitilde{}|≤k(b-a)\textasciicircum{}5\textasciicircum{}/180n\textasciicircum{}4\textasciicircum{}, k=f\textasciicircum{}(4)\textasciicircum{}(x)\textasciitilde{}max\textasciitilde{} on {[}a,b{]} \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Integral Approximations are typically used to evaluate an integral that is very difficult or impossible to integrate \newline  \newline Δx=(b-a)/n \newline  \newline x̄=(x\textasciitilde{}i-1\textasciitilde{}+x\textasciitilde{}i\textasciitilde{})/2, the average/median of two points x\textasciitilde{}i-1\textasciitilde{} and x\textasciitilde{}i\textasciitilde{} \newline  \newline Simpson's Rule can only be used if the given n is even, that is, n=2k for some integer k. \newline  \newline In order of most accurate to least accurate approximation: Simpson's Rule, Midpoint Rule, Trapezoidal Rule, Left/Right endpoint approximation}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}