4.08e Mean value of function: using integral

7 The function f is defined by $$f ( x ) = \frac { 1 } { \sqrt { x } } \quad 4 \leq x \leq 7$$ Find the mean value of f over the interval \(4 \leq x \leq 7\) Give your answer in exact form.

2 Find the mean value of \(3 x ^ { 2 }\) over the interval \(1 \leq x \leq 3\) Circle your answer.
[0pt] [1 mark] $$8 \frac { 2 } { 3 } \quad 10 \quad 13 \quad 26$$

2
3 2 Find the mean value of the function \(\mathrm { f } ( x ) = 10 x ^ { 4 }\) between \(x = 0\) and \(x = a\) Circle your answer.
[0pt] [1 mark] \(10 a ^ { 3 }\) \(40 a ^ { 3 }\) \(2 a ^ { 4 }\) \(4 a ^ { 5 }\)

9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\). Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2$$ Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\).

10

1. By using the substitution \(u = 3 - 2 x\), or otherwise, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 4 x } { 3 - 2 x } \right) ^ { 2 } \mathrm {~d} x = 16 - 12 \ln 3\).
2. \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-4_595_588_927_817} The diagram shows the region \(R\), which is bounded by the curve \(y = \frac { 4 x } { 3 - 2 x }\), the \(y\)-axis and the line \(y = 4\). Find the exact volume generated when the region \(R\) is rotated completely around the \(x\)-axis. {www.cie.org.uk} after the live examination series. }

Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express \(\sin^6 \theta\) in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4] Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]

Find the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x^2(1 - x).$$ [7] Deduce the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x(1 - x)^2.$$ [2]

It is given that \(I_n = \int_{1}^{e} (\ln x)^n \mathrm{d}x\) for \(n \geqslant 0\). Show that $$I_n = (n - 1)[I_{n-2} - I_{n-1}] \text{ for } n \geqslant 2.$$ [6] 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]

Answer only one of the following two alternatives. EITHER The curve \(C\) is defined parametrically by $$x = 18t - t^2 \quad \text{and} \quad y = 8t^{\frac{1}{2}},$$ where \(0 < t \leqslant 4\).

1. Show that at all points of \(C\), $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{-3(9 + t)}{2t^2(9 - t)^3}.$$ [4]
2. Show that the mean value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) with respect to \(x\) over the interval \(0 < x \leqslant 56\) is \(\frac{3}{70}\). [4]
3. Find the area of the surface generated when \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, showing full working. [6]

OR Let \(I_n = \int_1^{\sqrt{2}} (x^2 - 1)^n \mathrm{d}x\).

1. Show that, for \(n \geqslant 1\), $$(2n + 1)I_n = \sqrt{2} - 2nI_{n-1}.$$ [5]
2. Using the substitution \(x = \sec \theta\), show that $$I_n = \int_0^{\frac{1}{4}\pi} \tan^{2n+1} \theta \sec \theta \, \mathrm{d}\theta.$$ [4]
3. Deduce the exact value of $$\int_0^{\frac{1}{4}\pi} \frac{\sin^7 \theta}{\cos^8 \theta} \, \mathrm{d}\theta.$$ [5]

The curve \(C\) has equation \(y = x^a\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis. [6]

The curve \(C\) has equation \(y = x^a\) for \(0 \leq x \leq 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis. [6]

The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\). The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\) [2 marks]

The function \(f(x) = x^2 - 1\) Find the mean value of \(f(x)\) from \(x = -0.5\) to \(x = 1.7\) Give your answer to three significant figures. Circle your answer. [1 mark] \(-0.521\) \quad \(-0.434\) \quad \(-0.237\) \quad \(0.786\)

The function f is defined by $$f(x) = x^2 \quad (x \in \mathbb{R})$$ Find the mean value of \(f(x)\) between \(x = 0\) and \(x = 2\) Circle your answer. [1 mark] \(\frac{2}{3}\) \(\frac{4}{3}\) \(\frac{8}{3}\) \(\frac{16}{3}\)

A curve is defined parametrically by the equations $$x = \frac{3}{2}t^3 + 5$$ $$y = t^{\frac{3}{2}} \quad (t \geq 0)$$ Show that the arc length of the curve from \(t = 0\) to \(t = 2\) is equal to 26 units. [5 marks]

The function \(f(x) = \cosh(ix)\) is defined over the domain \(\{x \in \mathbb{R} : -a\pi \leq x \leq a\pi\}\), where \(a\) is a positive integer. By considering the graph of \(y = [f(x)]^n\), find the mean value of \([f(x)]^n\), when \(n\) is an odd positive integer. Fully justify your answer. [3 marks]

In this question you must show detailed reasoning.

1. Given that \(y = \arctan x\), show that \(\frac{dy}{dx} = \frac{1}{1+x^2}\). [3]

Fig. 12 shows the curve \(y = \frac{1}{1+x^2}\). \includegraphics{figure_12}

1. Find, in exact form, the mean value of the function \(f(x) = \frac{1}{1+x^2}\) for \(-1 \leq x \leq 1\). [3]
2. The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = -1\) is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [7]

The function \(f\) is defined by $$f(x) = \frac{1}{\sqrt{x^2 + 4x + 3}}.$$

1. Find the mean value of the function \(f\) for \(0 \leqslant x \leqslant 2\), giving your answer correct to three decimal places. [5]
2. The region \(R\) is bounded by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 2\). Find the exact value of the volume of the solid generated when \(R\) is rotated through four right angles about the \(x\)-axis. [6]

Given that \(y = \arcsin x\), \(-1 \leqslant x < 1\),

1. show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\). [3]

Given that \(f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}\),

1. show that the mean value of \(f(x)\) over the interval \([0, \sqrt{2}]\), is $$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$ where \(A\) is a constant to be determined. [6]

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

In this question you must show detailed reasoning. An ant starts from a fixed point \(O\) and walks in a straight line for \(1.5\) s. Its velocity, \(v\) cms\(^{-1}\), can be modelled by \(v = \frac{1}{\sqrt{9-t^2}}\). By finding the mean value of \(v\) in \(0 \leq t \leq 1.5\), deduce the average velocity of the ant. [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]