Volume of revolution with hyperbolics

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = 5 \cosh x - 6 \sinh x$$ The curve crosses the \(x\)-axis at the point \(A\).

1. Find the exact value of the \(x\) coordinate of the point \(A\), giving your answer as a natural logarithm.
2. Show that $$( 5 \cosh x - 6 \sinh x ) ^ { 2 } \equiv a \cosh 2 x + b \sinh 2 x + c$$ where \(a , b\) and \(c\) are constants to be found. The finite region \(R\), bounded by the curve and the coordinate axes, is shown shaded in Figure 1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Use calculus to find the volume of the solid generated, giving your answer as an exact multiple of \(\pi\).

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1. Show that the curve with equation $$y = 3 \sinh x - 2 \cosh x$$ has no turning points.
   Show that the curve crosses the \(x\)-axis at \(x = \frac { 1 } { 2 } \ln 5\). Show that this is also the point at which the gradient of the curve has a stationary value.
2. Sketch the curve.
3. Express \(( 3 \sinh x - 2 \cosh x ) ^ { 2 }\) in terms of \(\sinh 2 x\) and \(\cosh 2 x\). Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through \(360 ^ { \circ }\) about the \(x\)-axis is $$\pi \left( 3 - \frac { 5 } { 4 } \ln 5 \right) .$$ Option 2: Investigation of curves \section*{This question requires the use of a graphical calculator.}

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1. Express \(17 \cosh x - 15 \sinh x\) in the form \(\mathrm { e } ^ { - \mathrm { x } } \left( \mathrm { ae } ^ { \mathrm { bx } } + \mathrm { c } \right)\) where \(a , b\) and \(c\) are integers to be determined. A function is defined by \(\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 17 \cosh x - 15 \sinh x } }\). The region bounded by the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\). \section*{(b) In this question you must show detailed reasoning.} Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k \pi \tan ^ { - 1 } q\) where \(k\) and \(q\) are rational numbers to be determined.

11. (a) Find the area of the region enclosed by the curve \(y = x \sinh x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).
(b) The region \(R\) is bounded by the curve \(y = \cosh 2 x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis.
(c) Using your answer to part (b), find the total volume of the solid generated by rotating the region bounded by the curve \(y = \cosh 2 x\) and the lines \(x = - 1\) and \(x = 1\).

1. Solutions based entirely on graphical or numerical methods are not acceptable.

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \operatorname { arsinh } x \quad x \geqslant 0$$ and the straight line with equation \(y = \beta\)
The line and the curve intersect at the point with coordinates \(( \alpha , \beta )\)
Given that \(\beta = \frac { 1 } { 2 } \ln 3\)

1. show that \(\alpha = \frac { 1 } { \sqrt { 3 } }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \operatorname { arsinh } x\), the \(y\)-axis and the line with equation \(y = \beta\) The region \(R\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
2. Use calculus to find the exact value of the volume of the solid generated.