4.08e Mean value of function: using integral

10 Let \(I _ { n } = \int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { n } x \mathrm {~d} x\), where \(n \geqslant 0\).

1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cot ^ { n + 1 } x \right)\), or otherwise, show that $$I _ { n + 2 } = \frac { 1 } { n + 1 } - I _ { n }$$ The curve \(C\) has equation \(y = \cot x\), for \(\frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Find, in an exact form, the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x = \frac { 1 } { 4 } \pi\) and the \(x\)-axis.

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 Let \(y = \mathrm { e } ^ { x }\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 2\). Show that the mean value of \(x\) with respect to \(y\) over the interval \(1 \leqslant y \leqslant \mathrm { e } ^ { 2 }\) is \(\frac { \mathrm { e } ^ { 2 } + 1 } { \mathrm { e } ^ { 2 } - 1 }\).

1 Given that $$y = x ^ { 2 } \sin x$$

1. show that the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) is \(\frac { 1 } { 2 } \pi\),
2. find the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

3 A finite region \(R\) in the \(x - y\) plane is bounded by the curve with equation \(y = \sqrt { } x - \frac { 1 } { \sqrt { } x }\), the \(x\)-axis between \(x = 1\) and \(x = 4\), and the line \(x = 4\). Find the exact value of the \(y\)-coordinate of the centroid of \(R\).

6 [In this question you may use, without proof, the formula \(\int \sec x \mathrm {~d} x = \ln ( \sec x + \tan x ) + \operatorname { const }\).]

1. Let \(y = \sec x\). Find the mean value of \(y\) with respect to \(x\) over the interval \(\frac { 1 } { 6 } \pi \leqslant x \leqslant \frac { 1 } { 3 } \pi\).
2. The curve \(C\) has equation \(y = - \ln ( \cos x )\), for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). Find the arc length of \(C\).

7 The curve \(C\) has equation \(y = \mathrm { e } ^ { - 2 x }\). Find, giving your answers correct to 3 significant figures,

1. the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) over the interval \(0 \leqslant x \leqslant 2\),
2. the coordinates of the centroid of the region bounded by \(C\), \(x = 0\), \(x = 2\) and \(y = 0\).

9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\).

1. Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2 .$$
2. Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\). [6]

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 The function f is defined by $$f ( x ) = 3 x ^ { 2 } \quad 1 \leq x \leq 5$$ 5

1. Find the mean value of f
   5
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { f } ( x ) + c \quad 1 \leq x \leq 5$$ The mean value of \(g\) is 40
   Calculate the value of the constant \(c\)

1 Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).

8

1. Using exponentials, show that \(\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1\).
2. By differentiating both sides of the identity in part (a) with respect to \(u\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\).
3. Use the substitution \(\mathrm { x } = \sinh ^ { 2 } \mathrm { u }\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form asinh \(^ { - 1 } \mathrm {~b} \sqrt { \mathrm { x } } + \mathrm { f } ( \mathrm { x } )\) where \(a\) and \(b\) are integers and \(\mathrm { f } ( x )\) is a function to be determined.
4. Hence determine the exact area of the region between the curve \(\mathrm { y } = \sqrt { \frac { \mathrm { x } } { \mathrm { x } + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(\mathrm { p } + \mathrm { q } \mid \mathrm { nr }\) where \(p , q\) and \(r\) are numbers to be determined.

6 A particle, \(P\), positioned at the origin, \(O\), is projected with a certain velocity along the \(x\)-axis. \(P\) is then acted on by a single force which varies in such a way that \(P\) moves backwards and forwards along the \(x\)-axis. When the time after projection is \(t\) seconds, the displacement of \(P\) from the origin is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~ms} ^ { - 1 }\). The motion of \(P\) is modelled using the differential equation \(\ddot { x } + \omega ^ { 2 } x = 0\), where \(\omega\) rads \(^ { - 1 }\) is a positive constant.

1. Write down the general solution of this differential equation. \(D\) is the point where \(x = d\) for some positive constant, \(d\). When \(P\) reaches \(D\) it comes to instantaneous rest.
2. Using the answer to part (a), determine expressions, in terms of \(\omega\), \(d\) and \(t\) only, for the following quantities
   The quantity \(z\) is defined by \(z = \frac { 1 } { v }\).
3. Using part (c), determine an expression for \(\mathrm { Z } _ { \mathrm { m } }\), the mean value of z with respect to the displacement, as \(P\) moves directly from \(O\) to \(D\). One measure of the validity of the model is consideration of the value of \(\mathrm { z } _ { \mathrm { m } }\). If \(\mathrm { z } _ { \mathrm { m } }\) exceeds 8 then the model is considered to be valid. The value of \(d\) is measured as 0.25 to 2 significant figures. The value of \(\omega\) is measured as \(0.75 \pm 0.02\).
4. Determine what can be inferred about the validity of the model from the given information.
5. Find, according to the model, the least possible value of the velocity with which \(P\) was initially projected. Give your answer to \(\mathbf { 2 }\) significant figures.

7 The points \(P \left( \frac { 1 } { 2 } , \frac { 13 } { 24 } \right)\) and \(Q \left( \frac { 3 } { 2 } , \frac { 31 } { 24 } \right)\) lie on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }\).
The area of the surface generated when arc \(P Q\) is rotated completely about the \(x\)-axis is denoted by \(A\).

1. Find the exact value of \(A\). Give your answer as a rational multiple of \(\pi\). Student X finds an approximation to \(A\) by modelling the arc \(P Q\) as the straight line segment \(P Q\), then rotating this line segment completely about the \(x\)-axis to form a surface.
2. Find the approximation to \(A\) obtained by student X . Give your answer as a rational multiple of \(\pi\). Student Y finds a second approximation to \(A\) by modelling the original curve as the line \(y = M\), where \(M\) is the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }\), then rotating this line completely about the \(x\)-axis to form a surface.
3. Find the approximation to \(A\) obtained by student Y . Give your answer correct to four decimal places.

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 \(S\).

4 In this question you must show detailed reasoning.
Determine the mean value of \(\frac { 1 } { 1 + 4 x ^ { 2 } }\) between \(x = - 1\) and \(x = 1\). Give your answer to 3 significant
figures. figures.

1. (a) Write \(x ^ { 2 } + 4 x - 5\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are integers.
   (b) Hence use a standard integral from the formula book to find

$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form \(A \ln B\) where \(A\) and \(B\) are constants in simplest form.

6. $$\mathrm { f } ( x ) = \frac { x + 2 } { x ^ { 2 } + 9 }$$

1. Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \ln \left( x ^ { 2 } + 9 \right) + B \arctan \left( \frac { x } { 3 } \right) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
2. Hence show that the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\) is $$\frac { 1 } { 6 } \ln 2 + \frac { 1 } { 18 } \pi$$
3. Use the answer to part (b) to find the mean value, over the interval \([ 0,3 ]\), of $$\mathrm { f } ( x ) + \ln k$$ where \(k\) is a positive constant, giving your answer in the form \(p + \frac { 1 } { 6 } \ln q\), where \(p\) and \(q\) are constants and \(q\) is in terms of \(k\).

3. $$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$

1. Using a substitution, that should be stated clearly, show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
2. Hence find, in exact form in terms of natural logarithms, the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\).

1. (a)

$$y = \tan ^ { - 1 } x$$ Assuming the derivative of \(\tan x\), prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }$$ $$\mathrm { f } ( x ) = x \tan ^ { - 1 } 4 x$$ (b) Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A x ^ { 2 } \tan ^ { - 1 } 4 x + B x + C \tan ^ { - 1 } 4 x + k$$ where \(k\) is an arbitrary constant and \(A , B\) and \(C\) are constants to be determined.
(c) Hence find, in exact form, the mean value of \(\mathrm { f } ( x )\) over the interval \(\left[ 0 , \frac { \sqrt { 3 } } { 4 } \right]\)

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross-section \(A B C D E F A\) of a vase together with measurements that have been taken from the vase. The horizontal cross-section between \(A B\) and \(F C\) is a circle with diameter 4 cm .
The base of the vase \(E D\) is horizontal and the point \(E\) is vertically below \(F\) and the point \(D\) is vertically below \(C\). Using these measurements, the curve \(C D\) is modelled by the parametric equations $$x = a + 3 \sin 2 t \quad y = b \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Determine the value of \(a\) and the value of \(b\) according to the model.
2. Using algebraic integration and showing all your working, determine, according to the model, the volume of the vase, giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\)
3. State a limitation of the model.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 3 \cos \left( \frac { \pi } { 2 } x ^ { 2 } \right) \quad x \in [ - 1,1 ]$$ A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.
Use Simpson's rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in \(\mathrm { m } ^ { 3 }\) to 4 significant figures.

6 In this question you must show detailed reasoning. The power output, \(p\) watts, of a machine at time \(t\) hours after it is switched on can be modelled by the equation \(\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )\) for \(t \geqslant 0\). Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to \(\mathbf { 2 }\) decimal places.

2 Given that \(\mathrm { f } ( x ) = 3 x - 1\) find the mean value of \(\mathrm { f } ( x )\) over the interval \(4 \leq x \leq 8\) Circle your answer. 6111717