8.06b Arc length and surface area: of revolution, cartesian or parametric

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

1. Find the coefficient of \(x ^ { 2 }\) in the Maclaurin's series for \(- \ln \cos x\).
2. Find the length of the arc of the curve with equation \(\mathrm { y } = - \operatorname { Incos } \mathrm { x }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 4 } \pi\).

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 4 } { 3 } } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { - 1 }\) giving your answer in the form \(\ln a\), where \(a\) is an integer to be determined.
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm { n } \right)\), or otherwise, show that $$( \mathrm { n } + 1 ) \mathrm { I } _ { \mathrm { n } } = \mathrm { nl } _ { \mathrm { n } - 2 } + \frac { 4 } { 3 } \left( \frac { 5 } { 3 } \right) ^ { \mathrm { n } }$$
3. A curve has equation \(y = x ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The arc length of the curve is denoted by \(s\). Use the substitution \(\mathrm { u } = 2 \mathrm { x }\) to show that \(\mathrm { s } = \frac { 1 } { 2 } \mathrm { l } _ { 1 }\) and find the exact value of \(s\).

2 The curve \(C\) has parametric equations $$x = \cosh t , \quad y = \sinh t , \quad \text { for } 0 < t \leqslant \frac { 3 } { 5 }$$ The length of \(C\) is denoted by \(s\).

1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 5 } } \sqrt { \cosh 2 t } \mathrm {~d} t\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-04_2714_37_143_2008}
2. By finding the Maclaurin's series for \(\sqrt { \cosh 2 t }\) up to and including the term in \(t ^ { 2 }\) ,deduce an approximation to \(s\) .

3 The curve \(C\) has parametric equations $$\mathrm { x } = \mathrm { e } ^ { \mathrm { t } } - \frac { 1 } { 3 } \mathrm { t } ^ { 3 } , \quad \mathrm { y } = 4 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { t } } ( \mathrm { t } - 2 ) , \quad \text { for } 0 \leqslant t \leqslant 2$$ Find, in terms of e , the length of \(C\).

3 A curve has equation \(y = \mathrm { e } ^ { x }\) for \(\ln \frac { 4 } { 3 } \leqslant x \leqslant \ln \frac { 12 } { 5 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).

1. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that $$A = 2 \pi \int _ { \frac { 4 } { 3 } } ^ { \frac { 12 } { 5 } } \sqrt { 1 + u ^ { 2 } } \mathrm {~d} u$$
2. Use the substitution \(u = \sinh v\) to show that $$A = \pi \left( \frac { 904 } { 225 } + \ln \frac { 5 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-06_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-07_2726_35_97_20}

5 The curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 , \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$

1. Find, in terms of e , the length of \(C\).
2. Find, in terms of \(\pi\) and e , the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

1. The curve \(C\) has equation

$$y = 2 + \ln \left( 1 - x ^ { 2 } \right) \quad \frac { 1 } { 2 } \leqslant x \leqslant \frac { 3 } { 4 }$$

1. Show that the length of the curve \(C\) is given by $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x$$
2. Hence, using algebraic integration, show that the length of the curve \(C\) is \(p + \ln q\) where \(p\) and \(q\) are rational numbers to be determined.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( \sec \theta + \tan \theta ) - \sin \theta \quad y = \cos \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis and is used to form a solid of revolution \(S\). Using calculus, show that the total surface area of \(S\) is given by $$\frac { \pi } { 2 } ( p + q \sqrt { 2 } )$$ where \(p\) and \(q\) are integers to be determined.

1. A curve has parametric equations
   where \(a\) is a positive constant.

$$\begin{aligned} & x = a ( \theta - \sin \theta ) \\ & y = a ( 1 - \cos \theta ) \end{aligned}$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = k a ^ { 2 } \sin ^ { 2 } \frac { \theta } { 2 }$$ where \(k\) is a constant to be determined. The part of the curve from \(\theta = 0\) to \(\theta = 2 \pi\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Determine the area of the surface generated, giving your answer in terms of \(\pi\) and \(a\).
   [0pt] [Solutions relying on calculator technology are not acceptable.]

8. The curve \(C\) has parametric equations $$x = \theta - \sin \theta , \quad y = 1 - \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).

1. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { 2 \pi } ( 1 - \cos \theta ) ^ { \frac { 3 } { 2 } } \mathrm {~d} \theta$$
2. Hence find the exact value of \(S\).

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 \(\pi\).

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.

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.

1. A curve, which is part of an ellipse, has parametric equations

$$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { \alpha } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \quad \text { where } c = \cos \theta$$ and where \(k\) and \(\alpha\) are constants to be found.
2. Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.

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. 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)

4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.

3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).

1. Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Show that the equation of the normal to \(C\) at the point with parameter \(t\) is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
4. Find the cartesian equation of the envelope of the normals to \(C\).
5. The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).

3 The curve \(C\) has parametric equations \(x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }\), for \(t \geqslant 0\).

1. Show that \(\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }\). Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 2\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Show that the curvature at a general point on \(C\) is \(\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }\).
4. Find the coordinates of the centre of curvature corresponding to the point on \(C\) where \(t = 1\).

3 A curve \(C\) has equation \(y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } }\), for \(x \geqslant 0\).

1. Show that the arc of \(C\) for which \(0 \leqslant x \leqslant a\) has length \(a ^ { \frac { 1 } { 2 } } + \frac { 1 } { 3 } a ^ { \frac { 3 } { 2 } }\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Find the coordinates of the centre of curvature corresponding to the point \(\left( 4 , - \frac { 2 } { 3 } \right)\) on \(C\). The curve \(C\) is one member of the family of curves defined by $$y = p ^ { 2 } x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } p ^ { 3 } x ^ { \frac { 3 } { 2 } } \quad ( \text { for } x \geqslant 0 )$$ where \(p\) is a parameter (and \(p > 0\) ).
4. Find the equation of the envelope of this family of curves.

3 A curve has parametric equations $$x = a \left( 1 - \cos ^ { 3 } \theta \right) , \quad y = a \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ where \(a\) is a positive constant.
The arc length from the origin to a general point on the curve is denoted by \(s\), and \(\psi\) is the acute angle defined by \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).

1. Express \(s\) and \(\psi\) in terms of \(\theta\), and hence show that the intrinsic equation of the curve is $$s = \frac { 3 } { 2 } a \sin ^ { 2 } \psi$$
2. For the point on the curve given by \(\theta = \frac { \pi } { 6 }\), find the radius of curvature and the coordinates of the centre of curvature.
3. Find the area of the curved surface generated when the curve is rotated through \(2 \pi\) radians about the \(y\)-axis.

3

1. Find the length of the arc of the polar curve \(r = a ( 1 + \cos \theta )\) for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
2. A curve \(C\) has cartesian equation \(y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x }\).
   1. The arc of \(C\) for which \(1 \leqslant x \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a surface of revolution. Find the area of this surface. For the point on \(C\) at which \(x = 2\),
   2. show that the radius of curvature is \(\frac { 289 } { 64 }\),
   3. find the coordinates of the centre of curvature.

3

1. A curve has intrinsic equation \(s = 2 \ln \left( \frac { \pi } { \pi - 3 \psi } \right)\) for \(0 \leqslant \psi < \frac { 1 } { 3 } \pi\), where \(s\) is the arc length measured from a fixed point P and \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x } . \mathrm { P }\) is in the third quadrant. The curve passes through the origin O , at which point \(\psi = \frac { 1 } { 6 } \pi . \mathrm { Q }\) is the point on the curve at which \(\psi = \frac { 3 } { 10 } \pi\).
   1. Express \(\psi\) in terms of \(s\), and sketch the curve, indicating the points \(\mathrm { O } , \mathrm { P }\) and Q .
   2. Find the arc length OQ .
   3. Find the radius of curvature at the point O .
   4. Find the coordinates of the centre of curvature corresponding to the point O .
2. 1. Find the surface area of revolution formed when the curve \(y = \frac { 1 } { 3 } \sqrt { x } ( x - 3 )\) for \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
   2. The curve in part (b)(i) is one member of the family \(y = \frac { 1 } { 9 } \lambda \sqrt { x } ( x - \lambda )\), where \(\lambda\) is a positive parameter. Find the equation of the envelope of this family of curves.

3 A curve has parametric equations \(x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Show that the arc length \(s\) from the origin to a general point on the curve is given by \(s = 4 a \sin \frac { 1 } { 2 } \theta\).
2. Find the intrinsic equation of the curve giving \(s\) in terms of \(a\) and \(\psi\), where \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Hence, or otherwise, show that the radius of curvature at a point on the curve is \(4 a \cos \frac { 1 } { 2 } \theta\).
4. Find the coordinates of the centre of curvature corresponding to the point on the curve where \(\theta = \frac { 2 } { 3 } \pi\).
5. Find the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.