The end \(A\) of an inextensible string \(AB\) of length \(\pi\)
is attached to a point on the circumference
of a fixed circle of unit radius and
centre \(O\). Initially the string is straight
and tangent to the circle. The string is then wrapped round the circle
until the end \(B\) comes into
contact with the circle.
The string remains taut during the motion,
so that a section of the string is in contact with the circumference
and the remaining section is straight.
Taking \(O\) to be the origin of cartesian coordinates with \(A\) at \((-1,0)\)
and \(B\) initially at \((-1, \pi)\), show that the
curve described by \(B\) is given parametrically by
\[
x= \cos t + t\sin t\,, \ \ \ \ \ \
y= \sin t - t\cos t\,,
\]
where \(t\) is the angle shown in the diagram.
\psset{xunit=0.8cm,yunit=0.8cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3pt 0,linewidth=0.3pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-5.4,-1)(7,7)
\pspolygon(-1.22,3.03)(-0.87,3.17)(-1.01,3.52)(-1.36,3.38)
\parametricplot{-0.17}{3.3}{1*3.64*cos(t)+0*3.64*sin(t)+0|0*3.64*cos(t)+1*3.64*sin(t)+0}
\psline(-1.36,3.38)(6.23,6.37)
\psline[linestyle=dashed,dash=1pt 1pt](0,0)(-1.36,3.38)
\parametricplot{-0.0}{1.9540453733056695}{1.06*cos(t)+0|1.03*sin(t)+0}
\rput[tl](-0.45,-0.1){\(O\)}
\rput[tl](-4.12,0.46){\(A\)}
\rput[tl](6.11,6.8){\(B\)}
\rput[tl](0.25,0.6){\(t\)}
\psline{->}(-7.22,0)(5.78,0)
\psline{->}(0,-1.53)(0,6)
\rput[tl](-0.08,6.45){\(y\)}
\rput[tl](5.85,0.1){\(x\)}
\end{pspicture*}
Find the value, \(t_0\), of \(t\) for which \(x\)
takes its maximum value on the curve,
and sketch the curve.
Use the area integral $\displaystyle \int y \frac{\d x}{\d t} \,
\d t\,$
to find the area between the curve and the \(x\) axis
for~\hbox{\(\pi \ge t \ge t_0\)}.
Find the area swept out by the string (that is, the area between the
curve described by
\(B\) and the semicircle shown in the diagram).