\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
\mbox{(1a)} \qquad x
 & = & a \cos^{3} t \qquad - \pi \leq t \leq \pi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(1b)} \qquad y
 & = & a \sin^{3} t \qquad - \pi \leq t \leq \pi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad y
 & = & a \sin^{3} t \\
0
 & = & a \sin^{3} t \\
& = & \sin t \\
t
 & = & - \pi, \, \, \, 0, \, \, \, \pi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad x
 & = & a \cos^{3} t \\
& = & a \cos^{3} (- \pi) \, \, \, \vee \, \, \, a \cos^{3} 0 \, \, \, \vee \, \, \, a \cos^{3} \pi \\
& = & a (-1) \, \, \, \vee \, \, \, a (1) \, \, \, \vee \, \, \, a (-1) \\
& = & - a \, \, \, \vee \, \, \, a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad x
 & = & a \cos^{3} t \\
0
 & = & a \cos^{3} t \\
& = & \cos t \\
t
 & = & - \frac{1}{2} \pi, \, \, \, \frac{1}{2} \pi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad y
 & = & a \sin^{3} t \\
& = & a \sin^{3} \left ( - \frac{1}{2} \pi \right ) \, \, \, \vee \, \, \, a \sin^{3} \left ( \frac{1}{2} \pi \right ) \\
& = & a (-1) \, \, \, \vee \, \, \, a (1) \\
& = & - a \, \, \, \vee \, \, \, a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad a \sin^{3} t
 & = & a \cos^{3} t \\
\sin^{3} t
 & = & \cos^{3} t \\
\sin t
 & = & \cos t \\
t
 & = & - \frac{3}{4} \pi, \, \, \, - \frac{1}{4} \pi, \, \, \, \frac{1}{4} \pi, \, \, \, \frac{3}{4} \pi \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad x
 & = & a \cos^{3} t \\
& = & a \cos^{3} \left ( - \frac{3}{4} \pi \right ), \, \, \, a \cos^{3} \left ( - \frac{1}{4} \pi \right ),
 \, \, \, a \cos^{3} \left ( \frac{1}{4} \pi \right ), \, \, \, a \cos^{3} \left ( \frac{3}{4} \pi \right ) \\
& = & a \left ( - \frac{1}{2} \sqrt{2} \right )^{3}, \, \, \, a \left ( \frac{1}{2} \sqrt{2} \right )^{3} \\
& = & - a \left ( \frac{1}{4} \sqrt{2} \right ), \, \, \, a \left ( \frac{1}{4} \sqrt{2} \right ) \\
& = & - \frac{1}{4} a \sqrt{2}, \, \, \, \frac{1}{4} a \sqrt{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad r
 & = & \sqrt{x^{2} + y^{2}} \\
& = & \sqrt{\left ( \frac{1}{4} a \sqrt{2} \right )^{2} + \left ( \frac{1}{4} a \sqrt{2} \right )^{2}} \\
& = & \sqrt{2 \left ( \frac{1}{4} a \sqrt{2} \right )^{2}} \\
& = & \frac{1}{4} a \sqrt{2} \sqrt{2} \\
& = & \frac{1}{2} a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9a)} \qquad \frac{\mbox{d}x}{\mbox{d}t}
 & = & \frac{\mbox{d}}{\mbox{d}t} (a \cos^{3} t) \\
& = & - 3 a \cos^{2} t \sin t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9b)} \qquad \frac{\mbox{d}y}{\mbox{d}t}
 & = & \frac{\mbox{d}}{\mbox{d}t} (a \sin^{3} t) \\
& = & 3 a \sin^{2} t \cos t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad s
 & = & \int \sqrt{\left ( \frac{\mbox{d}x}{\mbox{d}t} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}t} \right )^{2}} \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad s
 & = & \int \sqrt{\left ( \frac{\mbox{d}x}{\mbox{d}t} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}t} \right )^{2}} \, \mbox{d}t \\
& = & \int \sqrt{(- 3 a \cos^{2} t \sin t)^{2} + (3 a \sin^{2} t \cos t)^{2}} \, \mbox{d}t \\
& = & \int \sqrt{9 a^{2} \cos^{4} t \sin^{2} t + 9 a^{2} \sin^{4} t \cos^{2} t} \, \mbox{d}t \\
& = & 3 a \int \cos t \sin t \sqrt{\cos^{2} t + \sin^{2} t} \, \mbox{d}t \\
& = & 3 a \int \cos t \sin t \sqrt{1} \, \mbox{d}t \\
& = & 3 a \int \cos t \sin t \, \mbox{d}t \\
& = & 3 a \cdot \frac{1}{2} \sin^{2} t \\
& = & \frac{3}{2} a \sin^{2} t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad s
 & = & \int\limits_{t = 0}^{t = \frac{1}{2} \pi}
 \sqrt{\left ( \frac{\mbox{d}x}{\mbox{d}t} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}t} \right )^{2}} \, \mbox{d}t \\
& = & \frac{3}{2} a \left [ \sin^{2} t \right ]_{t = 0}^{t = \frac{1}{2} \pi} \\
& = & \frac{3}{2} a \left ( \sin^{2} \left ( \frac{1}{2} \pi \right ) - \sin^{2} 0 \right ) \\
& = & \frac{3}{2} a (1 - 0) \\
& = & \frac{3}{2} a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad y
 & = & p x + q \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14)} \qquad y
 & = & p x + q \\
& = & \frac{\mbox{d}y}{\mbox{d}x} \, x + q \\
& = & \frac{\mbox{d}y / \mbox{d}t}{\mbox{d}x / \mbox{d}t} \, x + q \\
& = & \frac{3 a \sin^{2} t_{p} \cos t_{p}}{- 3 a \cos^{2} t_{p} \sin t_{p}} x + q \\
& = & - \frac{\sin t_{p}}{\cos t_{p}} x + q \\
& = & - \tan t_{p} \, x + q \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad y_{p}
 & = & - \tan t_{p} \, x_{p} + q \\
a \sin^{3} t_{p}
 & = & - \tan t_{p} \, a \cos^{3} t_{p} + q \\
q
 & = & a \sin^{3} t_{p} + a \tan t_{p} \, \cos^{3} t_{p} \\
& = & a \left ( \sin^{3} t_{p} + \frac{\sin t_{p}}{\cos t_{p}} \cos^{3} t_{p} \right ) \\
& = & a \sin t_{p} (\sin^{2} t_{p} + \cos^{2} t_{p}) \\
& = & a \sin t_{p} (1) \\
& = & a \sin t_{p} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad y
 & = & - \tan t_{p} \, x + q \\
& = & - \tan t_{p} \, x + a \sin t_{p} \\
& = & - \frac{\sin t_{p}}{\cos t_{p}} x + a \sin t_{p} \\
& = & \sin t_{p} \left ( - \frac{1}{\cos t_{p}} x + a \right ) \\
& = & \sin t_{p} (- \sec t_{p} \, x + a) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17a)} \qquad y
 & = & \sin t_{p} (- \sec t_{p} \, x + a) \\
0
 & = & \sin t_{p} (- \sec t_{p} \, x + a) \\
& = & - \sec t_{p} \, x + a \\
\sec t_{p} \, x
 & = & a \\
x
 & = & \frac{a}{\sec t_{p}} \\
& = & a \cos t_{p} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17b)} \qquad y
 & = & \sin t_{p} (- \sec t_{p} \, x + a) \\
& = & \sin t_{p} (- \sec t_{p} \dot 0 + a) \\
& = & \sin t_{p} (0 + a) \\
& = & \sin t_{p} (a) \\
& = & a \sin t_{p} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad r
 & = & \sqrt{x^{2} + y^{2}} \\
& = & \sqrt{(a \cos t_{p})^{2} + (a \sin t_{p})^{2}} \\
& = & \sqrt{a^{2} \cos^{2} t_{p} + a^{2} \sin^{2} t_{p}} \\
& = & \sqrt{a^{2} (\cos^{2} t_{p} + \sin^{2} t_{p})} \\
& = & \sqrt{a^{2} (1)} \\
& = & \sqrt{a^{2}} \\
& = & a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19a)} \qquad x
 & = & a \cos^{3} t \\
\cos^{3} t
 & = & \frac{x}{a} \\
\cos t
 & = & \sqrt[3]{\frac{x}{a}} \\
t
 & = & \arccos \sqrt[3]{\frac{x}{a}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19b)} \qquad y
 & = & a \sin^{3} t \\
& = & a (\sin t)^{3} \\
& = & a \left ( \sqrt{1 - \cos^{2} t} \, \right )^{3} \\
& = & a \left ( \sqrt{1 - \cos^{2} \arccos \sqrt[3]{\frac{x}{a}}} \, \right )^{3} \\
& = & a \left ( \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \, \right )^{3} \\
& = & a \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad \frac{\mbox{d}y}{\mbox{d}x}
 & = & \frac{\mbox{d}}{\mbox{d}x} \left ( a \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \right ) \\
& = & \frac{3}{2} a \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{1}{2}}
 \left ( - \frac{2}{3} \left ( \frac{1}{a} \right )^{\frac{2}{3}} x^{- \frac{1}{3}} \right ) \\
& = & - a^{\frac{1}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{1}{2}} x^{- \frac{1}{3}} \\
& = & - \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21)} \qquad y
 & = & p x + q \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22)} \qquad y
 & = & p x + q \\
& = & \frac{\mbox{d}y}{\mbox{d}x} \, x + q \\
& = & - \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x_{p}}} x + q \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23)} \qquad \qquad \qquad y_{p}
 & = & - \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x_{p}}} x_{p} + q \\
a \left ( 1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}}
 & = & - \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x_{p}}} x_{p} + q \\
q
 & = & a \left ( 1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}}
 + \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x_{p}}} x_{p} \\
& = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}}
 \left ( a \left ( 1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}} \right ) + \sqrt[3]{\frac{a}{x_{p}}} x_{p} \right ) \\
& = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}}
 \left ( a - a^{\frac{1}{3}} x_{p}^{\frac{2}{3}} + a^{\frac{1}{3}} x_{p}^{\frac{2}{3}} \right ) \\
& = & a \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24)} \qquad y
 & = & - \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x_{p}}} x + q \\
& = & - \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x_{p}}} x
 + a \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \\
& = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \left ( - \sqrt[3]{\frac{a}{x_{p}}} x + a \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25a)} \qquad y
 & = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \left ( - \sqrt[3]{\frac{a}{x_{p}}} x + a \right ) \\
0
 & = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \left ( - \sqrt[3]{\frac{a}{x_{p}}} x + a \right ) \\
& = & - \sqrt[3]{\frac{a}{x_{p}}} x + a \\
a
 & = & \sqrt[3]{\frac{a}{x_{p}}} x \\
x
 & = & a \left ( \sqrt[3]{\frac{a}{x_{p}}} \right )^{-1} \\
& = & a \sqrt[3]{\frac{x_{p}}{a}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25b)} \qquad y
 & = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \left ( - \sqrt[3]{\frac{a}{x_{p}}} x + a \right ) \\
& = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \left ( - \sqrt[3]{\frac{a}{x_{p}}} \cdot 0 + a \right ) \\
& = & \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \left ( 0 + a \right ) \\
& = & a \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26)} \qquad r
 & = & \sqrt{x^{2} + y^{2}} \\
& = & \sqrt{\left ( a \sqrt[3]{\frac{x_{p}}{a}} \, \right )^{2} + \left ( a \sqrt{1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \, \right )^{2}} \\
& = & \sqrt{a^{2} \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}} + a^{2} \left ( 1 - \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}} \right )} \\
& = & \sqrt{a^{2} \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}} + a^{2} - a^{2} \left ( \frac{x_{p}}{a} \right )^{\frac{2}{3}}} \\
& = & \sqrt{a^{2}} \\
& = & a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27)} \qquad A
 & = & \int\limits_{x = 0}^{x = a} a \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \, \mbox{d}x \\
& = & a \int\limits_{x = 0}^{x = a} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(28)} \qquad A
 & = & a \int\limits_{x = 0}^{x = a} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \, \mbox{d}x \\
& = & a \left [ \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right ) \right ]_{x = 0}^{x = a} \\
& = & \frac{1}{16} a^{2} \left (
 \left ( 3 \arcsin \sqrt[3]{\frac{a}{a}}
 + \sqrt[3]{\frac{a}{a}} \sqrt{1 - \left ( \frac{a}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{a}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{a}{a} \right )^{\frac{4}{3}} \right ) \right ) \right. \\
 & & \qquad \qquad \left . - \, \left ( 3 \arcsin \sqrt[3]{\frac{0}{a}}
 + \sqrt[3]{\frac{0}{a}} \sqrt{1 - \left ( \frac{0}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{0}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{0}{a} \right )^{\frac{4}{3}} \right ) \right ) \right ) \\
& = & \frac{1}{16} a^{2} ((3 \arcsin 1 + 1 \cdot \sqrt{1 - 1} (- 3 + 14 \cdot 1 - 8 \cdot 1)) \\
 & & \qquad \qquad - \, (3 \arcsin 0 + 0 \cdot \sqrt{1 - 0} (- 3 + 14 \cdot 0 - 8 \cdot 0))) \\
& = & \frac{1}{16} a^{2} \left ( 3 \cdot \frac{1}{2} \pi + 0 - 0 - 0 \right ) \\
& = & \frac{1}{16} a^{2} \frac{3}{2} \pi \\
& = & \frac{3}{32} \pi a^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(29)} \qquad A_{tot}
 & = & 4 A \\
& = & 4 \cdot \frac{3}{32} \pi a^{2} \\
& = & \frac{3}{8} \pi a^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30a)} \qquad \frac{\mbox{d}^{2}x}{\mbox{d}t^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{\mbox{d}x}{\mbox{d}t} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}t} (- 3 a \cos^{2} t \sin t) \\
& = & - 3 a (- 2 \cos t \sin^{2} t + \cos^{3} t) \\
& = & 3 a \cos t (2 \sin^{2} t - \cos^{2} t) \\
& = & 3 a \cos t (2 \sin^{2} t - (1 - \sin^{2} t)) \\
& = & 3 a \cos t (3 \sin^{2} t - 1) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30b)} \qquad \frac{\mbox{d}^{2}y}{\mbox{d}t^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{\mbox{d}y}{\mbox{d}t} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}t} (3 a \sin^{2} t \cos t) \\
& = & 3 a (2 \sin t \cos^{2} t - \sin^{3} t) \\
& = & 3 a \sin t (2 \cos^{2} t - \sin^{2} t) \\
& = & 3 a \sin t (2 \cos^{2} t - (1 - \cos^{2} t)) \\
& = & 3 a \sin t (3 \cos^{2} t - 1) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad \kappa
 & = & \frac{\frac{\mbox{d}^{2}y}{\mbox{d}t^{2}} \frac{\mbox{d}x}{\mbox{d}t} - \frac{\mbox{d}^{2}x}{\mbox{d}t^{2}} \frac{\mbox{d}y}{\mbox{d}t}}
{\left ( \left ( \frac{\mbox{d}x}{\mbox{d}t} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}t} \right )^{2} \right )^{3/2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad \kappa
 & = & \frac{\frac{\mbox{d}^{2}y}{\mbox{d}t^{2}} \frac{\mbox{d}x}{\mbox{d}t} - \frac{\mbox{d}^{2}x}{\mbox{d}t^{2}} \frac{\mbox{d}y}{\mbox{d}t}}
{\left ( \left ( \frac{\mbox{d}x}{\mbox{d}t} \right )^{2} + \left ( \frac{\mbox{d}y}{\mbox{d}t} \right )^{2} \right )^{3/2}} \\
& = & \frac{3 a \sin t (3 \cos^{2} t - 1) (- 3 a \cos^{2} t \sin t) - 3 a \cos t (3 \sin^{2} t - 1) (3 a \sin^{2} t \cos t)}
{\left ( \left ( - 3 a \cos^{2} t \sin t \right )^{2} + \left ( 3 a \sin^{2} t \cos t \right )^{2} \right )^{3/2}} \\
& = & \frac{- 9 a^{2} \cos^{2} t \sin^{2} t (3 \cos^{2} t - 1) - 9 a^{2} \sin^{2} t \cos^{2} t (3 \sin^{2} t - 1)}
{(9 a^{2} \cos^{4} t \sin^{2} t + 9 a^{2} \sin^{4} t \cos^{2} t)^{3/2}} \\
& = & - 9 a^{2} \cos^{2} t \sin^{2} t \frac{3 \cos^{2} t - 1 + 3 \sin^{2} t - 1}
{(9 a^{2} \cos^{2} t \sin^{2} t (\cos^{4} t + \sin^{2} t))^{3/2}} \\
& = & - 9 a^{2} \cos^{2} t \sin^{2} t \frac{3 (\cos^{2} t + \sin^{2} t) - 2}{(9 a^{2} \cos^{2} t \sin^{2} t (1))^{3/2}} \\
& = & - 9 a^{2} \cos^{2} t \sin^{2} t \frac{3 (1) - 2}{(9 a^{2} \cos^{2} t \sin^{2} t)^{3/2}} \\
& = & - 9 a^{2} \cos^{2} t \sin^{2} t \frac{3 - 2}{27 a^{3} \cos^{3} t \sin^{3} t} \\
& = & - \frac{1}{3 a \cos t \sin t} \\
& = & - \frac{2}{3 a \sin (2 t)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33)} \qquad \rho
 & = & \left | \, \frac{1}{\kappa} \, \right | \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(34)} \qquad \rho
 & = & \left | \, \frac{1}{\kappa} \, \right | \\
& = & \left | \, \frac{1}{\frac{2}{3 a \sin (2 t)}} \, \right | \\
& = & \left | \, \frac{3}{2} a \sin (2 t) \, \right | \\
& = & \frac{3}{2} a \, | \, \sin (2 t) \, | \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(35)} \qquad \kappa
 & = & \left ( 1 + \left ( \frac{\mbox{d}y}{\mbox{d}x} \right )^{2} \right )^{- \frac{3}{2}} \frac{\mbox{d}^{2}y}{\mbox{d}x^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(36)} \qquad \frac{\mbox{d}^{2}y}{\mbox{d}x^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{\mbox{d}y}{\mbox{d}x} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( - \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \, \sqrt[3]{\frac{a}{x}} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( - \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \right ) \\
& = & - \frac{1}{2} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}}
 \left ( 0 - \frac{2}{3} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \frac{1}{a} \right ) \left ( \frac{x}{a} \right )^{- \frac{1}{3}}
 - \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \left ( - \frac{1}{3} \right ) \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \frac{1}{a} \\
& = & \frac{1}{3a} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}}
 \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + \frac{1}{3a} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \\
& = & \frac{1}{3a} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}}
 \left ( \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right ) \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \right ) \\
& = & \frac{1}{3a} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}}
 \left ( \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + \left ( \frac{x}{a} \right )^{- \frac{4}{3}} - \left ( \frac{x}{a} \right )^{- \frac{2}{3}} \right ) \\
& = & \frac{1}{3a} \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(37)} \qquad \kappa
 & = & \left ( 1 + \left ( \frac{\mbox{d}y}{\mbox{d}x} \right )^{2} \right )^{- \frac{3}{2}} \frac{\mbox{d}^{2}y}{\mbox{d}x^{2}} \\
& = & \left ( 1 + \left ( - \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \right )^{2} \right )^{- \frac{3}{2}}
 \frac{1}{3a} \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
& = & \frac{1}{3a} \left ( 1 + \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )
 \left ( \frac{x}{a} \right )^{- \frac{2}{3}} \right )^{- \frac{3}{2}}
 \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
& = & \frac{1}{3a} \left ( 1 + \left ( \frac{x}{a} \right )^{- \frac{2}{3}} - 1 \right )^{- \frac{3}{2}}
 \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
& = & \frac{1}{3a} \left ( \left ( \frac{x}{a} \right )^{- \frac{2}{3}} \right )^{- \frac{3}{2}}
 \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
& = & \frac{1}{3a} \frac{x}{a}
 \left ( \frac{x}{a} \right )^{- \frac{4}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
& = & \frac{1}{3a} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(38)} \qquad \rho
 & = & \left | \, \frac{1}{\kappa} \, \right | \\
& = & \left | \, \left ( \frac{1}{3a} \left ( \frac{x}{a} \right )^{- \frac{1}{3}}
 \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \right )^{-1} \, \right | \\
& = & \left | \, 3a \left ( \frac{x}{a} \right )^{\frac{1}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{1}{2}} \, \right | \\
& = & 3a \sqrt[3]{\left | \, \frac{x}{a} \, \right |} \sqrt{1 - \left | \, \frac{x}{a} \, \right |^{\frac{2}{3}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(39)} \qquad \kappa
 & = & - \frac{2}{3 a \sin (2 t)} \\
& = & - \frac{1}{3 a \cos t \sin t} \\
& = & - (3 a \cos t \sin t)^{-1} \\
& = & - \left ( 3 a \cos t \sqrt{1 - \cos^{2} t} \, \right )^{-1} \\
& = & - \left ( 3 a \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \sqrt[3]{\frac{x}{a}} \right )^{2}} \, \right )^{-1} \\
& = & - \left ( 3 a \left ( \frac{x}{a} \right )^{\frac{1}{3}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \, \right )^{-1} \\
& = & - \frac{1}{3a} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
d_{1}
 & = & 2 a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
d_{2}
 & = & a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
O
 & = & 6 a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
A
 & = & \frac{3}{8} \pi a^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\kappa
 & = & \frac{1}{3a} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{- \frac{1}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\rho
 & = & 3a \sqrt[3]{\left | \, \frac{x}{a} \, \right |} \sqrt{1 - \left | \, \frac{x}{a} \, \right |^{\frac{2}{3}}} \\
\end{eqnarray*}
\end{document}
