Surface area of revolution: Cartesian curve

3

1. A curve has equation \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } + \frac { 1 } { 4 } \mathrm { e } ^ { - \mathrm { x } }\), for \(0 \leqslant x \leqslant 1\). 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.
2. Using standard results from the list of formulae (MF19), or otherwise, find the Maclaurin's series for \(\mathrm { e } ^ { x } + \frac { 1 } { 4 } \mathrm { e } ^ { - x }\) up to and including the term in \(x ^ { 2 }\).

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]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-06_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-07_2726_35_97_20}

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}

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.

4 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\). Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\). Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]

The curve \(C\) has equation $$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$ where \(\lambda > 0\) and \(0 \leqslant x \leqslant 3\). The length of \(C\) is denoted by \(s\). Prove that \(s = 2 \sqrt { } 3\). The area of the surface generated when \(C\) is rotated through one revolution about the \(x\)-axis is denoted by \(S\). Find \(S\) in terms of \(\lambda\). The \(y\)-coordinate of the centroid of the region bounded by \(C\), the axes and the line \(x = 3\) is denoted by h. Given that \(\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }\), show that $$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$

2 A curve has equation \(\mathrm { y } = \sqrt { 1 + \mathrm { x } ^ { 2 } }\), for \(0 \leqslant x \leqslant 1\), where both the \(x\) - and \(y\)-units are in cm. The area of the surface generated when this curve is rotated fully about the \(x\)-axis is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(\mathrm { A } = 2 \pi \int _ { 0 } ^ { 1 } \sqrt { 1 + \mathrm { kx } ^ { 2 } } \mathrm { dx }\) for some integer \(k\) to be determined. A small component for a car is produced in the shape of this surface. The curved surface area of the component must be \(8 \mathrm {~cm} ^ { 2 }\), accurate to within one percent. The engineering process produces such components with a curved surface area accurate to within one half of one percent.
2. Determine whether all components produced will be suitable for use in the car.

3 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\).

1. Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { 1 + x ^ { 4 } } \mathrm {~d} x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\).
2. Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { 2 } - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]

A curve has equation $$y = \sqrt{9 - x^2} \quad 0 \leq x \leq 3$$

1. Using calculus, show that the length of the curve is \(\frac{3\pi}{2}\) [4]

The curve is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Using calculus, find the exact area of the surface generated. [3]

The curve \(C\) has equation \(y = 2x^3\), \(0 \leq x \leq 2\). The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, find the area of the surface generated, giving your answer to 3 significant figures. [5]

A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x^2 + y^2 = r^2\) where \(r\) is a constant.

1. Show that \(1 + \left(\frac{dy}{dx}\right)^2 = \frac{r^2}{r^2 - x^2}\) [3]
2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4\pi r^2\). [5]
3. Write down the length of the arc of the curve \(y = \sqrt{1 - x^2}\) from \(x = 0\) to \(x = 1\) [1]

Let \(I_n = \int_0^2 x^n \sqrt{1 + 2x^2} \, \text{d}x\) for \(n = 0, 1, 2, 3, \ldots\).

1. 1. Evaluate \(I_1\). [3]
   2. Prove that, for \(n \geqslant 2\), $$(2n + 4)I_n = 27 \times 2^{n-1} - (n - 1)I_{n-2}.$$ [6]
   3. Using a suitable substitution, or otherwise, show that $$I_0 = 3 + \frac{1}{\sqrt{2}} \ln(1 + \sqrt{2}).$$ [8]
2. The curve \(y = \frac{1}{\sqrt{2}} x^2\), between \(x = 0\) and \(x = 2\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface with area \(S\). Find the exact value of \(S\). [5]

The curve \(C\) is given by \(y = \frac{1}{4}x^2 - \frac{1}{2}\ln x\) for \(2 \leq x \leq 8\).

1. Find, in its simplest exact form, the length of \(C\). [5]
2. When \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed. Show that the area of this surface is \(\pi(270 - 47\ln 2 - 2(\ln 2)^2)\). [10]