From 6b0f2df581e1357328cf53bde0a4d011517422d8 Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 10 Oct 2019 14:56:02 +1100
Subject: [PATCH] [spec] add section on circle mensuration

---
 spec/spec-collated.tex | 62 ++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 60 insertions(+), 2 deletions(-)

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 06cb974..38165bb 100644
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -818,7 +818,51 @@
                       \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
                     \end{axis}
                   \end{tikzpicture}
-\columnbreak
+
+                  \subsection*{Mensuration}
+
+                  \begin{tikzpicture}[draw=blue!70,thick]
+                    \filldraw[fill=lblue] circle (2cm);
+                    \filldraw[fill=white] 
+                    (320:2cm) node[right] {} 
+                    -- (220:2cm) node[left] {} 
+                    arc[start angle=220, end angle=320, radius=2cm] 
+                    -- cycle;
+                    \node {Major Segment};
+                    \node at (-90:2) {Minor Segment};
+
+                    \begin{scope}[xshift=4.5cm]
+                      \draw circle (2cm);
+                      \filldraw[fill=lblue] 
+                      (320:2cm) node[right] {}
+                      -- (0,0) node[above] {}
+                      -- (220:2cm) node[left] {} 
+                      arc[start angle=220, end angle=320, radius=2cm]
+                      -- cycle;
+                      \node at (90:1cm) {Major Sector};
+                      \node at (-90:1.5) {Minor Sector};
+                    \end{scope}
+                  \end{tikzpicture}
+
+                  \subsubsection*{Sectors}
+
+                  \begin{align*}
+                    A &= \pi r^2 \dfrac{\theta}{2\pi} \\
+                    &= \dfrac{r^2 \theta}{2}
+                  \end{align*}
+
+                  \subsubsection*{Segments}
+
+                  \[ A = \dfrac{r^2}{2} \left( \theta = \sin \theta \right) \]
+
+                  \subsubsection*{Chords}
+
+                  \begin{align*}
+                    \operatorname{crd} \theta &= \sqrt{(1 - \cos\theta)^2 + \sin^2 \theta} \\
+                    &= \sqrt{2 - 2\cos\theta} \\
+                    &= 2 \sin \left(\dfrac{\theta}{2}\right)
+                  \end{align*}
+
                   \section{Differential calculus}
 
                   \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
@@ -1068,6 +1112,10 @@
                     To reverse, use \texttt{combine(...)}
                   \end{cas}
 
+                  \subsection*{Integrating \(\boldsymbol{\dfrac{dy}{dx} = g(y)}\)}
+
+                  \[ \text{if } \dfrac{dy}{dx} = g(y), \text{ then } x = \int{\dfrac{1}{g(y)}} \> dy \]
+
                   \subsection*{Graphing integrals on CAS}
 
                   \begin{cas}
@@ -1166,10 +1214,20 @@
 
                   \subsection*{Euler's method}
 
-                  \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
+                  \[\dfrac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
 
                   \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
 
+                  \begin{theorembox}{}
+                    If \(\dfrac{dy}{dx} = g(x)\) with \(x_0 = a\) and \(y_0 = b\), then:
+                    \begin{align*}
+                      x_{n+1} &= x_n + h \\
+                      y_{n+1} &= y_n + hg(x_n)
+                    \end{align*}
+                  \end{theorembox}
+
+
+
                   \include{calculus-rules}
 
     \section{Kinematics \& Mechanics}
-- 
2.47.1