Surface area of revolution with hyperbolics

8 The curve \(C\) has parametric equations $$\mathbf { x } = 2 \cosh t , \quad \mathbf { y } = \frac { 3 } { 2 } \mathbf { t } - \frac { 1 } { 4 } \sinh 2 \mathbf { t } , \text { for } 0 \leqslant t \leqslant 1$$

1. Find \(\frac { \mathrm { dx } } { \mathrm { dt } }\) and show that \(\frac { \mathrm { dy } } { \mathrm { dt } } = 1 - \sinh ^ { 2 } \mathrm { t }\).
   The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
2. 1. Show that \(\mathrm { A } = \pi \int _ { 0 } ^ { 1 } \left( \frac { 3 } { 2 } \mathrm { t } - \frac { 1 } { 4 } \sinh 2 \mathrm { t } \right) ( 1 + \cosh 2 \mathrm { t } ) \mathrm { dt }\).
   2. Hence find \(A\) in terms of \(\pi , \sinh 2\) and \(\cosh 2\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A curve has equation \(\mathrm { y } = \cosh \mathrm { x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\).
Find, in terms of \(\pi\) and e, the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

8

1. Starting from the definition of cosh in terms of exponentials, prove that $$2 \cosh ^ { 2 } A = \cosh 2 A + 1$$ The curve \(C\) has parametric equations $$\mathrm { x } = 2 \cosh 2 \mathrm { t } + 3 \mathrm { t } , \quad \mathrm { y } = \frac { 3 } { 2 } \cosh 2 \mathrm { t } - 4 \mathrm { t } , \quad \text { for } - \frac { 1 } { 2 } \leqslant t \leqslant \frac { 1 } { 2 }$$ The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis is denoted by \(A\).
2. 1. Show that \(A = 10 \pi \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } ( 2 \cosh 2 t + 3 t ) \cosh 2 t d t\).
   2. Hence find \(A\) in terms of \(\pi\) and e.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6. The curve \(C\) has parametric equations $$x = \theta - \tanh \theta , \quad y = \operatorname { sech } \theta , \quad 0 \leqslant \theta \leqslant \ln 3$$

1. Find
   1. \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\)
   2. \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\) The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the exact area of the curved surface formed, giving your answer as a multiple of \(\pi\).

2. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = 2 \cosh \left( \frac { 1 } { 2 } x \right)\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates \(- \ln 2\) and \(\ln 2\) respectively. The arc of the curve joining \(A\) and \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the exact area of the curved surface area formed.
(Total 7 marks)

7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.

3. The curve with parametric equations $$x = \cosh 2 \theta , \quad y = 4 \sinh \theta , \quad 0 \leqslant \theta \leqslant 1$$ is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the surface generated is \(\lambda \left( \cosh ^ { 3 } \alpha - 1 \right)\), where \(\alpha = 1\) and \(\lambda\) is a constant to be found.

3

1. 1. Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\). The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
   2. Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
   3. Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Show that the normal to the ellipse at the point with parameter \(\theta\) has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
   2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
   3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
      (A) at the point \(( 2,0 )\),
      (B) at the point \(( 0,1 )\).

9 The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant \ln 5\). Find

1. the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \ln 5\),
2. the arc length of \(C\),
3. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

5 A designer intends to manufacture a product using a 3-D printer. The product will take the form of a surface \(S\) which must meet a number of design specifications. The designer chooses to model \(S\) with the equation \(\mathrm { Z } = \mathrm { y } \cosh \mathrm { x }\) for \(- \ln 20 \leqslant x \leqslant \ln 20 , - 2 \leqslant y \leqslant 2\).

1. 1. In the Printed Answer Booklet, on the axes provided, sketch the section of \(S\) given by \(y = 1\).
   2. One of the design specifications of the product is that this section should have a length no greater than 20 units. Determine whether the product meets this requirement according to the model.
2. 1. In the Printed Answer Booklet, on the axes provided, sketch the contour of \(S\) given by \(z = 1\).
   2. When this contour is rotated through \(2 \pi\) radians about the \(x\)-axis, the surface \(T\) is generated. The surface area of \(T\) is denoted by \(A\). Show that \(A\) can be written in the form \(\mathrm { k } \pi \int _ { 0 } ^ { \ln 20 } \frac { 1 } { \cosh ^ { 3 } \mathrm { x } } \sqrt { \cosh ^ { 4 } \mathrm { x } + \cosh ^ { 2 } \mathrm { x } - 1 } \mathrm { dx }\) for some
      integer \(k\) to be determined. integer \(k\) to be determined.
   3. A second design specification is that the surface area of \(T\) must not be greater than 20 square units. Use your calculator to decide whether the product meets this requirement according to the model.

4 A curve \(C\) is given parametrically by the equations $$x = \frac { 1 } { 2 } \cosh 2 t , \quad y = 2 \sinh t$$

1. Express $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 }$$ in terms of \(\cosh t\).
2. The arc of \(C\) from \(t = 0\) to \(t = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that \(S\), the area of the curved surface generated, is given by $$S = 8 \pi \int _ { 0 } ^ { 1 } \sinh t \cosh ^ { 2 } t \mathrm {~d} t$$
   2. Find the exact value of \(S\).

5

1. Use the definition \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) to show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
   (2 marks)
2. 1. The arc of the curve \(y = \cosh x\) between \(x = 0\) and \(x = \ln a\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that \(S\), the surface area generated, is given by $$S = 2 \pi \int _ { 0 } ^ { \ln a } \cosh ^ { 2 } x \mathrm {~d} x$$
   2. Hence show that $$S = \pi \left( \ln a + \frac { a ^ { 4 } - 1 } { 4 a ^ { 2 } } \right)$$

6

1. Show that $$\frac { 1 } { 4 } ( \cosh 4 x + 2 \cosh 2 x + 1 ) = \cosh ^ { 2 } x \cosh 2 x$$
2. Show that, if \(y = \cosh ^ { 2 } x\), then $$1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \cosh ^ { 2 } 2 x$$
3. The arc of the curve \(y = \cosh ^ { 2 } x\) between the points where \(x = 0\) and \(x = \ln 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = \frac { \pi } { 256 } ( a \ln 2 + b )$$ where \(a\) and \(b\) are integers.

7

1. 1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} u } \left( 2 u \sqrt { 1 + 4 u ^ { 2 } } + \sinh ^ { - 1 } 2 u \right) = k \sqrt { 1 + 4 u ^ { 2 } }$$ where \(k\) is an integer.
   2. Hence show that $$\int _ { 0 } ^ { 1 } \sqrt { 1 + 4 u ^ { 2 } } \mathrm {~d} u = p \sqrt { 5 } + q \sinh ^ { - 1 } 2$$ where \(p\) and \(q\) are rational numbers.
2. The arc of the curve with equation \(y = \frac { 1 } { 2 } \cos 4 x\) between the points where \(x = 0\) and \(x = \frac { \pi } { 8 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that the area \(S\) of the curved surface formed is given by $$S = \pi \int _ { 0 } ^ { \frac { \pi } { 8 } } \cos 4 x \sqrt { 1 + 4 \sin ^ { 2 } 4 x } \mathrm {~d} x$$
   2. Use the substitution \(u = \sin 4 x\) to find the exact value of \(S\).

10. \begin{figure}[h] \end{figure} A solid playing piece for a board game is modelled by rotating the curve \(C\), shown in Figure 2, through \(2 \pi\) radians about the \(x\)-axis. The curve \(C\) has equation $$y = \sqrt { 1 + \frac { x ^ { 2 } } { 9 } } \quad - 4 \leqslant x \leqslant 4$$ with units as centimetres.

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the playing piece is given by $$S = p \pi \int _ { - 4 } ^ { 4 } \sqrt { 81 + 10 x ^ { 2 } } \mathrm {~d} x + q \pi$$ where \(p\) and \(q\) are constants to be determined. Using the substitution \(x = \frac { 9 } { \sqrt { 10 } } \sinh u\), or another algebraic integration method, and showing all your working,
2. determine the total surface area of the playing piece, giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\)

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a French horn with a detachable bell section.
The shape of the bell section can be modelled by rotating an exponential curve through \(360 ^ { \circ }\) about the \(x\)-axis, where units are centimetres. The model uses the curve shown in Figure 2, with equation $$y = \frac { 9 } { 2 } e ^ { \frac { 1 } { 9 } x } \quad 0 \leqslant x \leqslant 9$$

1. Show that, according to this model, the external surface area of the bell section is given by $$K \int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x$$ where \(K\) is a real constant to be determined.
2. Use the substitution \(u = e ^ { \frac { 1 } { 9 } x }\) to show that $$\int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x = 9 \int _ { a } ^ { b } \frac { 2 u + u ^ { 3 } } { \sqrt { 4 u ^ { 2 } + u ^ { 4 } } } \mathrm {~d} u + 18 \int _ { a } ^ { b } \frac { 1 } { \sqrt { 4 + u ^ { 2 } } } \mathrm {~d} u$$ where \(a\) and \(b\) are constants to be determined. Hence, using algebraic integration,
3. determine, according to the model, the external surface area of the bell section of the horn, giving your answer to 3 significant figures.