\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \qquad (x - a)^{2} + (y - b)^{2}
 & = & R^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad (x - a)^{2} + (y - b)^{2}
 & = & R^{2} \\
(y - b)^{2}
 & = & R^{2} - (x - a)^{2} \\
y - b
 & = & \sqrt{R^{2} - (x - a)^{2}} \\
y
 & = & \sqrt{R^{2} - (x - a)^{2}} + b \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad y
 & = & \pm \sqrt{R^{2} - (x - a)^{2}} + b \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad s
 & = & \int \sqrt{1 + \left ( \frac{\mbox{d}y}{\mbox{d}x} \right )^{2}} \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad \frac{\mbox{d}y}{\mbox{d}x}
 & = & \frac{\mbox{d}}{\mbox{d}x} (\sqrt{R^{2} - (x - a)^{2}} + b) \\
& = & \frac{1}{2} \frac{1}{\sqrt{R^{2} - (x - a)^{2}}} (0 - 2 (x - a) \cdot 1) + 0 \\
& = & \frac{1}{2} \frac{1}{\sqrt{R^{2} - (x - a)^{2}}} (- 2 (x - a)) \\
& = & - \frac{x - a}{\sqrt{R^{2} - (x - a)^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad s
 & = & \int \sqrt{1 + \left ( \frac{\mbox{d}y}{\mbox{d}x} \right )^{2}} \, \mbox{d}x \\
& = & \int \sqrt{1 + \left ( - \frac{x - a}{\sqrt{R^{2} - (x - a)^{2}}} \right )^{2}} \, \mbox{d}x \\
& = & \int \sqrt{1 + \frac{(x - a)^{2}}{R^{2} - (x - a)^{2}}} \, \mbox{d}x \\
& = & \int \sqrt{\frac{R^{2} - (x - a)^{2}}{R^{2} - (x - a)^{2}} + \frac{(x - a)^{2}}{R^{2} - (x - a)^{2}}} \, \mbox{d}x \\
& = & \int \sqrt{\frac{R^{2} - (x - a)^{2} + (x - a)^{2}}{R^{2} - (x - a)^{2}}} \, \mbox{d}x \\
& = & \int \sqrt{\frac{R^{2}}{R^{2} - (x - a)^{2}}} \, \mbox{d}x \\
& = & R \int \sqrt{\frac{1}{R^{2} - (x - a)^{2}}} \, \mbox{d}x \\
& = & R \int \frac{1}{\sqrt{R^{2} - (x - a)^{2}}} \, \mbox{d} (x - a) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad s
 & = & R \int \frac{1}{\sqrt{R^{2} - (x - a)^{2}}} \, \mbox{d} (x - a) \\
& = & R \arcsin \frac{x - a}{R} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad s
 & = & \left [ R \arcsin \frac{x - a}{R} \right ]_{x = a}^{x = a + R} \\
& = & R \arcsin \frac{a + R - a}{R} - R \arcsin \frac{a - a}{R} \\
& = & R \arcsin \frac{R}{R} - R \arcsin \frac{0}{R} \\
& = & R \arcsin 1 - R \arcsin 0 \\
& = & R \frac{\pi}{2} - 0 \\
& = & \frac{1}{2} \pi R \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9)} \qquad O
 & = & 4 \cdot \frac{1}{2} \pi R \\
& = & 2 \pi R \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad y
 & = & \pm \sqrt{R^{2} - (x - a)^{2}} + b \\
& = & \pm \sqrt{R^{2} - (x - 0)^{2}} + R \\
& = & \pm \sqrt{R^{2} - x^{2}} + R \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad \frac{\epsilon}{2 \pi R}
 & = & \frac{\varphi}{2 \pi} \\
\varphi
 & = & \frac{\epsilon}{R} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12a)} \qquad \sin \varphi
 & = & \frac{\epsilon - x}{R} \\
\epsilon - x
 & = & R \sin \varphi \\
x
 & = & \epsilon - R \sin \varphi \\
& = & \epsilon - R \sin \frac{\epsilon}{R} \\
& = & R \left ( \frac{\epsilon}{R} - \sin \frac{\epsilon}{R} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12b)} \qquad \cos \varphi
 & = & \frac{R - y}{R} \\
R - y
 & = & R \cos \varphi \\
y
 & = & R - R \cos \varphi \\
& = & R - R \cos \frac{\epsilon}{R} \\
& = & R \left ( 1 - \cos \frac{\epsilon}{R} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13a)} \qquad x
 & = & R \left ( \frac{\epsilon}{R} - \sin \frac{\epsilon}{R} \right ) \\
& = & R (\varphi - \sin \varphi) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13b)} \qquad y
 & = & R \left ( 1 - \cos \frac{\epsilon}{R} \right ) \\
& = & R (1 - \cos \varphi) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14a)} \qquad \frac{\mbox{d}x}{\mbox{d}\varphi}
 & = & \frac{\mbox{d}}{\mbox{d}\varphi} (R (\varphi - \sin \varphi)) \\
& = & R (1 - \cos \varphi) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14b)} \qquad \frac{\mbox{d}y}{\mbox{d}\varphi}
 & = & \frac{\mbox{d}}{\mbox{d}\varphi} (R (1 - \cos \varphi)) \\
& = & R (0 + \sin \varphi) \\
& = & R \sin \varphi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad s
 & = & \int \sqrt{\left ( \frac{\mbox{d}x}{\mbox{d}\varphi} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}\varphi} \right )^{2}} \, \mbox{d}\varphi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad s
 & = & \int \sqrt{\left ( \frac{\mbox{d}x}{\mbox{d}\varphi} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}\varphi} \right )^{2}} \, \mbox{d}\varphi \\
& = & \int \sqrt{(R (1 - \cos \varphi))^{2} + (R \sin \varphi)^{2}} \, \mbox{d}\varphi \\
& = & \int \sqrt{R^{2} (1 - 2 \cos \varphi + \cos^{2} \varphi) + R^{2} \sin^{2} \varphi} \, \mbox{d}\varphi \\
& = & R \int \sqrt{1 - 2 \cos \varphi + \cos^{2} \varphi + \sin^{2} \varphi} \, \mbox{d}\varphi \\
& = & R \int \sqrt{1 - 2 \cos \varphi + 1} \, \mbox{d}\varphi \\
& = & R \int \sqrt{2 - 2 \cos \varphi} \, \mbox{d}\varphi \\
& = & \sqrt{2} R \int \sqrt{1 - \cos \varphi} \, \mbox{d}\varphi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17)} \qquad s
 & = & \sqrt{2} R \int \sqrt{1 - \cos \varphi} \, \mbox{d}\varphi \\
& = & - \sqrt{2} R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}}
 \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}}
 \left ( x + \sin x \sum_{i = 0}^{i = \frac{1}{2} n - 1} \frac{2^{2i} (i!)^{2}}{(2i + 1)!} \cos^{2i + 1} x \right ) \\
 n \mbox{ = oneven: }
 \sum_{i = 0}^{i = \frac{1}{2} (n - 1)} \left ( {\frac{1}{2} (n - 1) \atop i} \right ) \frac{(-1)^{i}}{2 i + 1} \sin^{2 i + 1} x \\
 \end{array}} \right . \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad s
 & = & \left [ - \sqrt{2} R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \right . \\
 & & \qquad \qquad \qquad \left . \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}}
 \left ( x + \sin x \sum_{i = 0}^{i = \frac{1}{2} n - 1} \frac{2^{2i} (i!)^{2}}{(2i + 1)!} \cos^{2i + 1} x \right ) \\
 n \mbox{ = oneven: }
 \sum_{i = 0}^{i = \frac{1}{2} (n - 1)} \left ( {\frac{1}{2} (n - 1) \atop i} \right ) \frac{(-1)^{i}}{2 i + 1} \sin^{2 i + 1} x \\
 \end{array}} \right . \right ]_{\varphi = 0}^{\varphi = 2 \pi} \\
& = & \left ( - \sqrt{2} R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \right . \\
 & & \qquad \qquad \qquad \left . \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}}
 \left ( 2 \pi + \sin (2 \pi) \sum_{i = 0}^{i = \frac{1}{2} n - 1} \frac{2^{2i} (i!)^{2}}{(2i + 1)!} \cos^{2i + 1} (2 \pi) \right ) \\
 n \mbox{ = oneven: }
 \sum_{i = 0}^{i = \frac{1}{2} (n - 1)} \left ( {\frac{1}{2} (n - 1) \atop i} \right ) \frac{(-1)^{i}}{2 i + 1} \sin^{2 i + 1} (2 \pi) \\
 \end{array}} \right . \right ) \\
 & & \qquad - \, \left ( - \sqrt{2} R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \right . \\
 & & \qquad \qquad \qquad \qquad \left . \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}}
 \left ( 0 + \sin 0 \sum_{i = 0}^{i = \frac{1}{2} n - 1} \frac{2^{2i} (i!)^{2}}{(2i + 1)!} \cos^{2i + 1} 0 \right ) \\
 n \mbox{ = oneven: }
 \sum_{i = 0}^{i = \frac{1}{2} (n - 1)} \left ( {\frac{1}{2} (n - 1) \atop i} \right ) \frac{(-1)^{i}}{2 i + 1} \sin^{2 i + 1} 0 \\
 \end{array}} \right . \right ) \\
& = & \left ( - \sqrt{2} R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}}
 \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}} (2 \pi + 0) \\
 n \mbox{ = oneven: } 0 \\
 \end{array}} \right . \right ) \\
 & & \qquad - \, \left ( - \sqrt{2} R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}}
 \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: } 0 \\
 n \mbox{ = oneven: } 0 \\
 \end{array}} \right . \right ) \\
& = & - 2 \sqrt{2} \pi R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}}
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}} \qquad \qquad n \mbox{ = even} \\
& = & - 2 \sqrt{2} \pi R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{3n} n!}
 \frac{1}{\left ( \left ( \frac{1}{2} n \right )! \right )^{2}} \qquad \qquad n \mbox{ = even} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19)} \qquad s
 & = & - 2 \sqrt{2} \pi R \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{3n} n!}
 \frac{1}{\left ( \left ( \frac{1}{2} n \right )! \right )^{2}} \qquad \qquad n \mbox{ = even} \\
& = & - 2 \sqrt{2} \pi R \sum_{2 k = 0}^{2 k = \infty} \frac{(2 \cdot 2 k)!}{(2 \cdot 2 k - 1) 2^{3 \cdot 2 k} (2 k)!}
 \frac{1}{\left ( \left ( \frac{1}{2} \cdot 2 k \right )! \right )^{2}} \qquad \qquad 2 k \mbox{ = even} \\
& = & - 2 \sqrt{2} \pi R \sum_{k = 0}^{k = \infty} \frac{(4 k)!}{(4 k - 1) 2^{6 k} (2 k)! (k!)^{2}} \\
& = & R \left ( - 2 \sqrt{2} \pi \sum_{k = 0}^{k = \infty} \frac{(4 k)!}{(4 k - 1) 2^{6 k} (2 k)! (k!)^{2}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad \frac{\mbox{Term } (k + 1)}{\mbox{Term } (k)}
 & = & \frac{\frac{(4 (k + 1))!}{(4 (k + 1) - 1) 2^{6 (k + 1)} (2 (k + 1))! ((k + 1)!)^{2}}}{\frac{(4 k)!}{(4 k - 1) 2^{6 k} (2 k)! (k!)^{2}}} \\
& = & \frac{\frac{(4k + 4)!}{(4k + 3) 2^{6k + 6} (2k + 2)! ((k + 1)!)^{2}}}{\frac{(4 k)!}{(4 k - 1) 2^{6 k} (2 k)! (k!)^{2}}} \\
& = & \frac{\frac{(4k)! (4k + 1) (4k + 2) (4k + 3) (4k + 4)}{(4k + 3) 2^{6k + 6} (2k)! (2k + 1) (2k + 2) (k! (k + 1))^{2}}}
{\frac{(4 k)!}{(4 k - 1) 2^{6 k} (2 k)! (k!)^{2}}} \\
& = & \frac{(4k + 1) (2k + 1) (k + 1) (4 k - 1)}{2^{4} (2k + 1) (k + 1) (k + 1) (k + 1)} \\
& = & \frac{(4k + 1) (4 k - 1)}{2^{4} (k + 1) (k + 1)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21)} \qquad s
 & = & 8 R \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22)} \qquad \frac{\mbox{d}y}{\mbox{d}x}
 & = & \frac{\frac{\mbox{d}y}{\mbox{d}\varphi}}{\frac{\mbox{d}x}{\mbox{d}\varphi}} \\
& = & \frac{R \sin \varphi}{R (1 - \cos \varphi)} \\
& = & \frac{\sin \varphi}{1 - \cos \varphi} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23)} \qquad \frac{\mbox{d}y}{\mbox{d}x}
 & = & 0 \\
\frac{\sin \varphi}{1 - \cos \varphi}
 & = & 0 \\
\sin \varphi
 & = & 0 \\
\varphi
 & = & 0 \quad \pmod{\pi} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24)} \qquad \frac{\mbox{d}y}{\mbox{d}x}
 & = & \infty \\
\frac{\sin \varphi}{1 - \cos \varphi}
 & = & \infty \\
1 - \cos \varphi
 & = & 0 \\
\cos \varphi
 & = & 1 \\
\varphi
 & = & 0 \quad \pmod{2 \pi} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25a)} \qquad x
 & = & R (\varphi - \sin \varphi) \\
& = & R (\varphi - \sqrt{1 - \cos^{2} \varphi} \, ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25b)} \qquad y
 & = & R (1 - \cos \varphi) \\
\frac{y}{R}
 & = & 1 - \cos \varphi \\
\cos \varphi
 & = & 1 - \frac{y}{R} \\
\varphi
 & = & \arccos \left ( 1 - \frac{y}{R} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26)} \qquad x
 & = & R (\varphi - \sqrt{1 - \cos^{2} \varphi} \, ) \\
& = & R \left ( \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{1 - \cos^{2} \arccos \left ( 1 - \frac{y}{R} \right )} \, \right ) \\
& = & R \left ( \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{1 - \left ( 1 - \frac{y}{R} \right )^{2}} \, \right ) \\
& = & R \left ( \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{1 - 1 + \frac{2 y}{R} - \frac{y^{2}}{R^{2}}} \, \right ) \\
& = & R \left ( \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{\frac{2 y}{R} - \frac{y^{2}}{R^{2}}} \, \right ) \\
& = & R \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{2 R y - y^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27)} \qquad X
 & = & \int \left ( R \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{2 R y - y^{2}} \, \right ) \, \mbox{d}y \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(28)} \qquad X
 & = & \int \left ( R \arccos \left ( 1 - \frac{y}{R} \right ) - \sqrt{2 R y - y^{2}} \, \right ) \, \mbox{d}y \\
& = & R \left ( - R \left ( 1 - \frac{y}{R} \right ) \arccos \left ( 1 - \frac{y}{R} \right )
 + R \sqrt{1 - \left ( 1 - \frac{y}{R} \right )^{2}} \, \right ) \\
 & & \qquad \qquad
 + \, \frac{1}{8} \left ( (- 2y + 2R) \sqrt{(2R)^{2} - (- 2y + 2R)^{2}} + (2R)^{2} \arcsin \frac{- 2y + 2R}{2R} \right ) \\
& = & R \left ( (y - R) \arccos \left ( 1 - \frac{y}{R} \right )
 + R \sqrt{1 - \left ( 1 - \frac{2y}{R} + \frac{y^{2}}{R^{2}} \right )} \, \right ) \\
 & & \qquad \qquad
 + \, \frac{1}{8} \left ( 2 (R - y) \sqrt{4R^{2} - 4y^{2} + 8 R y - 4R^{2}} + 4R^{2} \arcsin \frac{R - y}{R} \right ) \\
& = & R (y - R) \arccos \left ( 1 - \frac{y}{R} \right ) + R \sqrt{2 R y - y^{2}}
 + \frac{1}{2} (R - y) \sqrt{2 R y - y^{2}} + \frac{1}{2} R^{2} \arcsin \left ( 1 - \frac{y}{R} \right ) \\
& = & \frac{1}{2} R^{2} \arcsin \left ( 1 - \frac{y}{R} \right ) + R (y - R) \arccos \left ( 1 - \frac{y}{R} \right )
 + \frac{1}{2} (3 R - y) \sqrt{2 R y - y^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(29)} \qquad A_{geel}
 & = & \left [ \frac{1}{2} R^{2} \arcsin \left ( 1 - \frac{y}{R} \right ) + R (y - R) \arccos \left ( 1 - \frac{y}{R} \right )
 + \frac{1}{2} (3 R - y) \sqrt{2 R y - y^{2}} \, \right ]_{y = 0}^{y = 2 R} \\
& = & \frac{1}{2} R^{2} \arcsin \left ( 1 - \frac{2 R}{R} \right ) + R (2 R - R) \arccos \left ( 1 - \frac{2 R}{R} \right )
 + \frac{1}{2} (3 R - 2 R) \sqrt{2 R (2 R) - (2 R)^{2}} \\
 & & \qquad - \, \left ( \frac{1}{2} R^{2} \arcsin \left ( 1 - \frac{0}{R} \right ) + R (0 - R) \arccos \left ( 1 - \frac{0}{R} \right )
 + \frac{1}{2} (3 R - 0) \sqrt{2 R \cdot 0 - 0^{2}} \, \right ) \\
& = & \frac{1}{2} R^{2} \arcsin (1 - 2) + R^{2} \arccos (1 - 2) + \frac{1}{2} R \sqrt{4 R^{2} - 4 R^{2}} \\
 & & \qquad - \, \left ( \frac{1}{2} R^{2} \arcsin (1 - 0) - R^{2} \arccos (1 - 0) + 0 \right ) \\
& = & \frac{1}{2} R^{2} \arcsin (-1) + R^{2} \arccos (-1) + 0 - \frac{1}{2} R^{2} \arcsin 1 + R^{2} \arccos 1 \\
& = & \frac{1}{2} R^{2} \frac{- \pi}{2} + R^{2} \pi - \frac{1}{2} R^{2} \frac{\pi}{2} + R^{2} \cdot 0 \\
& = & - \frac{1}{4} R^{2} \pi + R^{2} \pi - \frac{1}{4} R^{2} \pi \\
& = & \frac{1}{2} \pi R^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30)} \qquad A_{groen}
 & = & 2 R \cdot \pi R \\
& = & 2 \pi R^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad A_{blauw}
 & = & A_{groen} - A_{geel} \\
& = & 2 \pi R^{2} - \frac{1}{2} \pi R^{2} \\
& = & \frac{3}{2} \pi R^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad A
 & = & 2 A_{blauw} \\
& = & 2 \cdot \frac{3}{2} \pi R^{2} \\
& = & 3 \pi R^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33a)} \qquad \frac{\mbox{d}^{2}x}{\mbox{d}\varphi^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}\varphi} \left ( \frac{\mbox{d}x}{\mbox{d}\varphi} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}\varphi} (R (1 - \cos \varphi)) \\
& = & R (0 + \sin \varphi) \\
& = & R \sin \varphi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33b)} \qquad \frac{\mbox{d}^{2}y}{\mbox{d}\varphi^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}\varphi} \left ( \frac{\mbox{d}y}{\mbox{d}\varphi} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}\varphi} (R \sin \varphi) \\
& = & R \cos \varphi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(34)} \qquad \kappa
 & = & \frac{\frac{\mbox{d}^{2}y}{\mbox{d}\varphi^{2}} \frac{\mbox{d}x}{\mbox{d}\varphi}
 - \frac{\mbox{d}^{2}x}{\mbox{d}\varphi^{2}} \frac{\mbox{d}y}{\mbox{d}\varphi}}
{\left ( \left ( \frac{\mbox{d}x}{\mbox{d}\varphi} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}\varphi} \right )^{2} \right )^{3/2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(35)} \qquad \kappa
 & = & \frac{\frac{\mbox{d}^{2}y}{\mbox{d}\varphi^{2}} \frac{\mbox{d}x}{\mbox{d}\varphi}
 - \frac{\mbox{d}^{2}x}{\mbox{d}\varphi^{2}} \frac{\mbox{d}y}{\mbox{d}\varphi}}
{\left ( \left ( \frac{\mbox{d}x}{\mbox{d}\varphi} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}\varphi} \right )^{2} \right )^{3/2}} \\
& = & \frac{R \cos \varphi R (1 - \cos \varphi) - R \sin \varphi R \sin \varphi}{((R (1 - \cos \varphi))^{2} + (R \sin \varphi)^{2})^{3/2}} \\
& = & \frac{R^{2} \cos \varphi - R^{2} \cos^{2} \varphi - R^{2} \sin^{2} \varphi}
{(R^{2} - 2 R^{2} \cos \varphi + R^{2} \cos^{2} \varphi + R^{2} \sin^{2} \varphi)^{3/2}} \\
& = & \frac{R^{2} \cos \varphi - R^{2} (\cos^{2} \varphi + \sin^{2} \varphi)}{R^{3} (1 - 2 \cos \varphi + (\cos^{2} \varphi + \sin^{2} \varphi))^{3/2}} \\
& = & \frac{\cos \varphi - 1}{R (1 - 2 \cos \varphi + 1)^{3/2}} \\
& = & - \frac{1 - \cos \varphi}{R (2 - 2 \cos \varphi)^{3/2}} \\
& = & - \frac{1 - \cos \varphi}{2 \sqrt{2} R (1 - \cos \varphi)^{3/2}} \\
& = & - \frac{1}{2 \sqrt{2} R \sqrt{1 - \cos \varphi}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(36)} \qquad \rho
 & = & \left | \, \frac{1}{\kappa} \, \right | \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(37)} \qquad \rho
 & = & \left | \, \frac{1}{\kappa} \, \right | \\
& = & \left | \, \frac{1}{- \frac{1}{\sqrt{2} R \sqrt{1 - \cos \varphi}}} \, \right | \\
& = & \left | \, \sqrt{2} R \sqrt{1 - \cos \varphi} \, \right | \\
& = & \sqrt{2} R \sqrt{1 - \cos \varphi} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
s
 & = & 8 R \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
A
 & = & 3 \pi R^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\kappa
 & = & - \frac{1}{2 \sqrt{2} R \sqrt{1 - \cos \varphi}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\rho
 & = & \sqrt{2} R \sqrt{1 - \cos \varphi} \\
\end{eqnarray*}
\end{document}
