Arc length with hyperbolic curves

8

1. Sketch the graph of \(\mathrm { y } = \operatorname { coth } \mathrm { x }\) for \(x > 0\) and state the equations of the asymptotes.
2. Starting from the definitions of coth and cosech in terms of exponentials, prove that $$\operatorname { coth } ^ { 2 } x - \operatorname { cosech } ^ { 2 } x = 1$$ The curve \(C\) has equation \(\mathrm { y } = \ln \operatorname { coth } \left( \frac { 1 } { 2 } \mathrm { x } \right)\) for \(x > 0\).
3. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosechx }\).
4. It is given that the arc length of \(C\) from \(\mathrm { x } = \mathrm { a }\) to \(\mathrm { x } = 2 \mathrm { a }\) is \(\ln 4\), where \(a\) is a positive constant. Show that \(\cosh a = 2\) and find, in logarithmic form, the exact value of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y ^ { 2 } = 8 x\) and part of the line \(l\) with equation \(x = 18\) The region \(R\), shown shaded in Figure 2, is bounded by \(C\) and \(l\)

1. Show that the perimeter of \(R\) is given by $$\alpha + 2 \int _ { 0 } ^ { \beta } \sqrt { 1 + \frac { y ^ { 2 } } { 16 } } d y$$ where \(\alpha\) and \(\beta\) are positive constants to be determined.
2. Use the substitution \(y = 4 \sinh u\) and algebraic integration to determine the exact perimeter of \(R\), giving your answer in simplest form.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

\begin{figure}[h] \end{figure} Figure 1 shows the curve with equation $$y = \ln \left( \tanh \frac { x } { 2 } \right) \quad 1 \leqslant x \leqslant 2$$

1. Show that the length, \(s\), of the curve is given by $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
2. Hence show that $$s = \ln \left( \mathrm { e } + \frac { 1 } { \mathrm { e } } \right)$$

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } \operatorname { arcosh } ( 2 x ) \quad \frac { 7 } { 2 } \leqslant x \leqslant 13$$ Using calculus, determine the exact length of the curve \(C\).
Give your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are constants to be found.

2. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has equation $$y = \frac { 1 } { 3 } \cosh 3 x , \quad 0 \leqslant x \leqslant \ln a$$ where \(a\) is a constant and \(a > 1\) Using calculus, show that the length of curve \(C\) is $$k \left( a ^ { 3 } - \frac { 1 } { a ^ { 3 } } \right)$$ and state the value of the constant \(k\).

8. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 2, has equation $$y = 2 x ^ { \frac { 1 } { 2 } } , \quad 1 \leqslant x \leqslant 8$$

1. Show that the length \(s\) of curve \(C\) is given by the equation $$s = \int _ { 1 } ^ { 8 } \sqrt { } \left( 1 + \frac { 1 } { x } \right) \mathrm { d } x$$
2. Using the substitution \(x = \sinh ^ { 2 } u\), or otherwise, find an exact value for \(s\). Give your answer in the form \(a \sqrt { } 2 + \ln ( b + c \sqrt { } 2 )\) where \(a , b\) and \(c\) are integers.

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } \ln ( \operatorname { coth } x ) , \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosech } 2 x$$ The points \(A\) and \(B\) lie on \(C\). The \(x\) coordinates of \(A\) and \(B\) are \(\ln 2\) and \(\ln 3\) respectively.
2. Find the length of the arc \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
   (6)

2. A curve has equation $$y = \cosh x , \quad 1 \leqslant x \leqslant \ln 5$$ Find the exact length of this curve. Give your answer in terms of e .

8. The curve \(C\) has equation $$y = \ln \left( \frac { \mathrm { e } ^ { x } + 1 } { \mathrm { e } ^ { x } - 1 } \right) , \quad \ln 2 \leqslant x \leqslant \ln 3$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$$
2. Find the length of the curve \(C\), giving your answer in the form \(\ln a\), where \(a\) is a rational number.
   (6)

1 The curve \(C\) has equation \(y = \frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\). Show that the length of the \(\operatorname { arc }\) of \(C\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 2 }\) is \(\frac { \mathrm { e } ^ { 2 } - 1 } { 4 \mathrm { e } }\).

3 A curve has cartesian equation $$y = \frac { 1 } { 2 } \ln ( \tanh x )$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sinh 2 x }$$
2. The points \(A\) and \(B\) on the curve have \(x\)-coordinates \(\ln 2\) and \(\ln 4\) respectively. Find the arc length \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.

5. \begin{figure}[h] \end{figure} An engineering student makes a miniature arch as part of the design for a piece of coursework. The cross-section of this arch is modelled by the curve with equation $$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$ where \(a > 1\) and \(A\) is a positive constant. The curve begins and ends on the \(x\)-axis, as shown in Figure 1.

1. Show that the length of this curve is \(k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)\), stating the value of the constant \(k\). The length of the curved cross-section of the miniature arch is required to be 2 m long.
2. Find the height of the arch, according to this model, giving your answer to 2 significant figures.
3. Find also the width of the base of the arch giving your answer to 2 significant figures.
4. Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
   (You do not need to consider the arc length of your curve)

13. \begin{figure}[h] \end{figure} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh \left( \frac { x } { a } \right) , \quad - k a \leq x \leq k a$$ where \(a\) and \(k\) are positive constants.

1. Prove that the length of the rope is \(2 a \sinh k\). Given that the length of the rope is \(8 a\),
2. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate.

7

1. Use the definitions $$\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right) \quad \text { and } \quad \cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)$$ to show that:
   1. \(2 \sinh \theta \cosh \theta = \sinh 2 \theta\);
   2. \(\cosh ^ { 2 } \theta + \sinh ^ { 2 } \theta = \cosh 2 \theta\).
2. A curve is given parametrically by $$x = \cosh ^ { 3 } \theta , \quad y = \sinh ^ { 3 } \theta$$
   1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = \frac { 9 } { 4 } \sinh ^ { 2 } 2 \theta \cosh 2 \theta$$
   2. Show that the length of the arc of the curve from the point where \(\theta = 0\) to the point where \(\theta = 1\) is $$\frac { 1 } { 2 } \left[ ( \cosh 2 ) ^ { \frac { 3 } { 2 } } - 1 \right]$$

4

1. Given that \(y = \operatorname { sech } t\), show that:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - \operatorname { sech } t \tanh t\);
   2. \(\left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \operatorname { sech } ^ { 2 } t - \operatorname { sech } ^ { 4 } t\).
2. The diagram shows a sketch of part of the curve given parametrically by $$x = t - \tanh t \quad y = \operatorname { sech } t$$
   \includegraphics[max width=\textwidth, alt={}]{1891766e-7744-49ac-82b6-7e51cb63b381-3_424_625_863_703}
   The curve meets the \(y\)-axis at the point \(K\), and \(P ( x , y )\) is a general point on the curve. The arc length \(K P\) is denoted by \(s\). Show that:
   1. \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \tanh ^ { 2 } t\);
   2. \(s = \ln \cosh t\);
   3. \(y = \mathrm { e } ^ { - s }\).
3. The arc \(K P\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the surface area generated is $$2 \pi \left( 1 - \mathrm { e } ^ { - S } \right)$$ (4 marks)

7

1. Given that \(y = \ln \tanh \frac { x } { 2 }\), where \(x > 0\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x$$
2. A curve has equation \(y = \ln \tanh \frac { x } { 2 }\), where \(x > 0\). The length of the arc of the curve between the points where \(x = 1\) and \(x = 2\) is denoted by \(s\).
   1. Show that $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
   2. Hence show that \(s = \ln ( 2 \cosh 1 )\).

7

1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \cosh ^ { - 1 } \frac { 1 } { x } \right) = \frac { - 1 } { x \sqrt { 1 - x ^ { 2 } } }$$ (3 marks)
2. A curve has equation $$y = \sqrt { 1 - x ^ { 2 } } - \cosh ^ { - 1 } \frac { 1 } { x } \quad ( 0 < x < 1 )$$ Show that:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - x ^ { 2 } } } { x }\);
      (4 marks)
   2. the length of the arc of the curve from the point where \(x = \frac { 1 } { 4 }\) to the point where $$x = \frac { 3 } { 4 } \text { is } \ln 3 .$$ (5 marks)

5 A curve has equation \(y = \cosh x\) Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to
\(\sinh b - \sinh a\)
\(6 \quad\) A circle \(C\) in the complex plane has equation \(| z - 2 - 5 \mathrm { i } | = a\) The point \(z _ { 1 }\) on \(C\) has the least argument of any point on \(C\), and \(\arg \left( z _ { 1 } \right) = \frac { \pi } { 4 }\) Prove that \(a = \frac { 3 \sqrt { } 2 } { 2 }\)