Volume with trigonometric functions

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1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) d x$$
2. \includegraphics[max width=\textwidth, alt={}, center]{48ab71ff-c37b-4e0b-b031-d99b0cf517a8-3_550_785_1573_721} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.

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1. Show that the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 2 } x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x\) is \(\frac { 1 } { 6 } \pi + \frac { 9 } { 8 } \sqrt { } 3\).
2. \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_444_495_1523_865} The diagram shows the curve \(y = \cos x + \frac { 1 } { \cos x }\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). The shaded region is bounded by the curve and the lines \(x = 0 , x = \frac { 1 } { 3 } \pi\) and \(y = 0\). Find the exact volume of the solid obtained when the shaded region is rotated completely about the \(x\)-axis.

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = x ( \sin x + \cos x ) , \quad 0 \leqslant x \leqslant \frac { \pi } { 4 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\). This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution, with volume \(V\).

1. Assuming the formula for volume of revolution show that \(V = \int _ { 0 } ^ { \frac { \pi } { 4 } } \pi x ^ { 2 } ( 1 + \sin 2 x ) \mathrm { d } x\)
2. Hence using calculus find the exact value of \(V\). You must show your working.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

6. \begin{figure}[h] \end{figure} The curve shown in Figure 2 has equation $$y ^ { 2 } = 3 \tan \left( \frac { x } { 2 } \right) , \quad 0 < x < \pi , \quad y > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac { \pi } { 3 }\) the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to generate a solid of revolution.
Show that the exact value of the volume of the solid generated may be written as \(A \ln \left( \frac { 3 } { 2 } \right)\), where \(A\) is a constant to be found.

9. \begin{figure}[h] \end{figure} (c) Find the exact value for the volume of this solid, giving your answer as a single, simplified fraction. \section*{Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\).
The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\). The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.} \(\_\_\_\_\) simplified fraction.

3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.

3. The diagram shows the curve with equation \(y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi\).
The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 2 }\) is rotated through four right angles about the \(x\)-axis. Show that the volume of the solid formed is \(\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )\).

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1. Find \(\int ( x + \cos 2 x ) ^ { 2 } \mathrm {~d} x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{80f94db1-39be-46f5-896e-277c93cbe4b8-3_538_935_383_646} The diagram shows the part of the curve \(y = x + \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The shaded region bounded by the curve, the axes and the line \(x = \frac { 1 } { 2 } \pi\) is rotated completely about the \(x\)-axis to form a solid of revolution of volume \(V\). Find \(V\), giving your answer in an exact form.

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1. Given that \(z = \frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \sec ^ { 2 } x\).
2. Sketch the curve with equation \(y = \sec x\) for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
3. The region \(R\) is bounded by the curve \(y = \sec x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi\). The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 2 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid formed is \(\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )\).
4. continued

12

1. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
2. By considering \(\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 }\), find constants \(A\) and \(B\) such that \(\sin ^ { 3 } \theta \cos ^ { 3 } \theta = A \sin 6 \theta + B \sin 2 \theta\).

1. (a) Given

$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad n \in \mathbb { N }$$ show that $$32 \cos ^ { 6 } \theta \equiv \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10$$ \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a solid paperweight with a flat base.
Figure 2 shows the curve with equation $$y = H \cos ^ { 3 } \left( \frac { x } { 4 } \right) \quad - 4 \leqslant x \leqslant 4$$ where \(H\) is a positive constant and \(x\) is in radians.
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = - 4\), the line with equation \(x = 4\) and the \(x\)-axis. The paperweight is modelled by the solid of revolution formed when \(R\) is rotated \(\mathbf { 1 8 0 } ^ { \circ }\) about the \(x\)-axis. Given that the maximum height of the paperweight is 2 cm ,
(b) write down the value of \(H\).
(c) Using algebraic integration and the result in part (a), determine, in \(\mathrm { cm } ^ { 3 }\), the volume of the paperweight, according to the model. Give your answer to 2 decimal places.
[0pt] [Solutions based entirely on calculator technology are not acceptable.]
(d) State a limitation of the model.

4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h] \end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.

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- 1 \end{array} \right) + \mu \left( \begin{array} { c } 1
- 2
4 \end{array} \right) .$$ (ii) Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).\\ (iii) Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). 2 In this question you must show detailed reasoning.\\ (i) Find \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 2 \tan x \mathrm {~d} x\) giving your answer in the form \(\ln p\).\\ (ii) Show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan x \mathrm {~d} x\) is undefined explaining your reasoning. 3 The equation of a plane, \(\Pi\), is $$\Pi : \quad \mathbf { r } = \left( \begin{array} { c } 2
- 3
5 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
3 \end{array} \right) + \mu \left( \begin{array} { c } - 1
2
1 \end{array} \right) .$$ (i) Find a vector which is perpendicular to \(\Pi\).\\ (ii) Hence find an equation for \(\Pi\) in the form r.n \(= p\).\\ (iii) Find in the form \(\sqrt { q }\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. 4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c r c } a & 2 & 3 \\ 4 & 4 & 6 \\ - 2 & 2 & 9 \end{array} \right)\) where \(a\) is a constant. It is given that if \(\mathbf { A }\) is not singular then $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 24 a - 48 } \left( \begin{array} { c c c } 24 & - 12 & 0
- 48 & 9 a + 6 & 12 - 6 a
16 & - 2 a - 4 & 4 a - 8 \end{array} \right)$$ (i) Use \(\mathbf { A } ^ { - 1 }\) to solve the simultaneous equations below, giving your answer in terms of \(k\). $$\begin{array} { r } x + 2 y + 3 z = 6
4 x + 4 y + 6 z = 8
- 2 x + 2 y + 9 z = k \end{array}$$ (ii) Consider the equations below where \(a\) takes the value which makes \(\mathbf { A }\) singular. $$\begin{aligned} a x + 2 y + 3 z & = b
4 x + 4 y + 6 z & = 10
- 2 x + 2 y + 9 z & = - 13 \end{aligned}$$ \(b\) takes the value for which the equations have an infinite number of solutions.

• Determine the value of \(b\).
• Find the solutions for \(y\) and \(z\) in terms of \(x\).
  (iii) For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations.

5 The region \(R\) between the \(x\)-axis, the curve \(y = \frac { 1 } { \sqrt { p + x ^ { 2 } } }\) and the lines \(x = \sqrt { p }\) and \(x = \sqrt { 3 p }\), where \(p\) is a positive parameter, is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
(i) Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\).
(ii) Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt { 48 }\) find in exact form

• the greatest possible value of the volume of \(S\)
• the least possible value of the volume of \(S\).

6 (i) By considering \(\sum _ { r = 1 } ^ { n } \left( ( r + 1 ) ^ { 5 } - r ^ { 5 } \right)\) show that \(\sum _ { r = 1 } ^ { n } r ^ { 4 } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( 3 n ^ { 2 } + 3 n - 1 \right)\).
(ii) Use the formula given in part (i) to find \(50 ^ { 4 } + 51 ^ { 4 } + \ldots + 80 ^ { 4 }\). 7 The roots of the equation \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are positive integers, are \(\alpha\) and \(\beta\).
(i) Find a quadratic equation with integer coefficients whose roots are \(\alpha + \beta\) and \(\alpha \beta\).
(ii) Show that it is not possible for the original equation and the equation found in part (i) both to have repeated roots.
(iii) Show that the discriminant of the equation found in part (i) is always positive.