4.08e Mean value of function: using integral

9 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689} The diagram shows the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\). The shaded region between the curve, the \(x\)-axis and the line \(x = \mathrm { e }\) is denoted by \(R\).

1. Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = m x + c\).
2. Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 2 } { 3 } \mathrm { t } ^ { \frac { 3 } { 2 } } - 2 \mathrm { t } ^ { \frac { 1 } { 2 } } , \quad \mathrm { y } = 2 \mathrm { t } + 5 , \quad \text { for } 0 < t \leqslant 3$$

1. Find the exact length of \(C\).
2. Find the set of values of \(t\) for which \(\frac { d ^ { 2 } y } { d x ^ { 2 } } > 0\).

7

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$2 \sinh ^ { 2 } A = \cosh 2 A - 1$$ \includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_79_1556_358_347} \includegraphics[max width=\textwidth, alt={}]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_69_1575_466_328} ....................................................................................................................................... ........................................................................................................................................
2. A curve has equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
   Use the substitution \(\mathrm { X } = \frac { 1 } { 2 } \operatorname { sinhu }\) to show that \(S = \frac { 1 } { 32 } \pi \left( \frac { 820 } { 81 } - \ln 3 \right)\).

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}

4. \begin{figure}[h] \end{figure} The curve \(C\) with equation \(y = \frac { 2 } { ( 4 + 3 x ) } , x > - \frac { 4 } { 3 }\) is shown in Figure 1
The region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = \frac { 2 } { 3 }\), is shown shaded in Figure 1 This region is rotated through 360 degrees about the \(x\)-axis.

1. Use calculus to find the exact value of the volume of the solid generated. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-05_583_433_1398_753} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a candle with axis of symmetry \(A B\) where \(A B = 15 \mathrm {~cm}\). \(A\) is a point at the centre of the top surface of the candle and \(B\) is a point at the centre of the base of the candle. The candle is geometrically similar to the solid generated in part (a).
2. Find the volume of this candle.

11. (a) Given $$\frac { 9 } { t ^ { 2 } ( t - 3 ) } \equiv \frac { A } { t } + \frac { B } { t ^ { 2 } } + \frac { C } { ( t - 3 ) }$$ find the value of the constants \(A , B\) and \(C\).
(b) $$I = \int _ { 4 } ^ { 12 } \frac { 9 } { t ^ { 2 } ( t - 3 ) } \mathrm { d } t$$ Find the exact value of \(I\), giving your answer in the form \(\ln ( a ) - b\), where \(a\) and \(b\) are positive constants. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 2 \ln ( t - 3 ) , \quad y = \frac { 6 } { t } \quad t > 3$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2 \ln 9\) The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
(c) Show that the exact volume of the solid generated is $$k \times I$$ where \(k\) is a constant to be found.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = x ^ { 3 } \ln \left( x ^ { 2 } + 2 \right) , x \geqslant 0\).
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = \sqrt { } 2\). The table below shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 3 } \ln \left( x ^ { 2 } + 2 \right)\).

1. Complete the table above giving the missing values of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. Use the substitution \(u = x ^ { 2 } + 2\) to show that the area of \(R\) is $$\frac { 1 } { 2 } \int _ { 2 } ^ { 4 } ( u - 2 ) \ln u \mathrm {~d} u$$
4. Hence, or otherwise, find the exact area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).

1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
2. Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. Question 7 continued
   8. (a) Find \(\int ( 4 y + 3 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} y\) (b) Given that \(y = 1.5\) at \(x = - 2\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { } ( 4 y + 3 ) } { x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$

1. Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
2. Show that the cartesian equation of \(C\) may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of \(a\) and \(b\).
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Use calculus to find the exact value of the volume of the solid of revolution. \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Let \(x\) be the mass of waste products, in kg , at time \(t\) minutes after the start of the experiment. It is known that at time \(t\) minutes, the rate of increase of the mass of waste products, in kg per minute, is \(k\) times the mass of unburned fuel remaining, where \(k\) is a positive constant. The differential equation connecting \(x\) and \(t\) may be written in the form $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
   1. Explain, in the context of the problem, what \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(M\) represent. Given that initially the mass of waste products is zero,
   2. solve the differential equation, expressing \(x\) in terms of \(k , M\) and \(t\). Given also that \(x = \frac { 1 } { 2 } M\) when \(t = \ln 4\),
   3. find the value of \(x\) when \(t = \ln 9\), expressing \(x\) in terms of \(M\), in its simplest form. \section*{Question 8 continued}

4

1. Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for \(\cosh 2 x\).
2. Sketch the curve with equation \(y = \cosh x - 1\). The region bounded by this curve, the \(x\)-axis, and the line \(x = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
3. Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
   Determine, in exact logarithmic form, the \(x\)-coordinate of the stationary point.

1

1. A curve has polar equation \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve and the line AB .
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
3. 1. Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
   2. Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).

1

1. Fig. 1 shows the curve with polar equation \(r = a ( 1 - \cos 2 \theta )\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-2_529_620_577_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Find the area of the region enclosed by the curve.
2. 1. Given that \(\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   2. Hence find the Maclaurin series for \(\arctan ( \sqrt { 3 } + x )\), as far as the term in \(x ^ { 2 }\).
   3. Hence show that, if \(h\) is small, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).

3 A curve has parametric equations \(x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Show that the arc length \(s\) from the origin to a general point on the curve is given by \(s = 4 a \sin \frac { 1 } { 2 } \theta\).
2. Find the intrinsic equation of the curve giving \(s\) in terms of \(a\) and \(\psi\), where \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Hence, or otherwise, show that the radius of curvature at a point on the curve is \(4 a \cos \frac { 1 } { 2 } \theta\).
4. Find the coordinates of the centre of curvature corresponding to the point on the curve where \(\theta = \frac { 2 } { 3 } \pi\).
5. Find the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

4

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.}

6 The curve \(C\) is defined parametrically by $$x = 4 t - t ^ { 2 } \quad \text { and } \quad y = 1 - \mathrm { e } ^ { - t }$$ where \(0 \leqslant t < 2\). Show that at all points of \(C\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { ( t - 1 ) \mathrm { e } ^ { - t } } { 4 ( 2 - t ) ^ { 3 } }$$ Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 7 } { 4 }\) is $$\frac { 4 e ^ { - \frac { 1 } { 2 } } - 3 } { 21 }$$

9 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ where \(n\) is a non-negative integer. Show that \(I _ { n + 2 } = \frac { n + 1 } { n + 2 } I _ { n }\). The region \(R\) of the \(x - y\) plane is bounded by the \(x\)-axis, the line \(x = \frac { \pi } { 2 m }\) and the curve whose equation is \(y = \sin ^ { 4 } m x\), where \(m > 0\). Find the \(y\)-coordinate of the centroid of \(R\).

9 The curve \(C\) has equation \(y = x ^ { \frac { 3 } { 2 } }\). Find the coordinates of the centroid of the region bounded by \(C\), the lines \(x = 1 , x = 4\) and the \(x\)-axis. Show that the length of the arc of \(C\) from the point where \(x = 5\) to the point where \(x = 28\) is 139 .

10 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Show that, for all \(n \geqslant 2\), $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$ A curve has parametric equations \(x = a \sin ^ { 3 } t\) and \(y = a \cos ^ { 3 } t\), where \(a\) is a constant and \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). Show that the mean value \(m\) of \(y\) over the interval \(0 \leqslant x \leqslant a\) is given by $$m = 3 a \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( \cos ^ { 4 } t - \cos ^ { 6 } t \right) \mathrm { d } t$$ Find the exact value of \(m\), in terms of \(a\).

10 The curve \(C\) has equation $$y = 2 \left( \frac { x } { 3 } \right) ^ { \frac { 3 } { 2 } }$$ where \(0 \leqslant x \leqslant 3\). Show that the arc length of \(C\) is \(2 ( 2 \sqrt { 2 } - 1 )\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 3\).

Show that $$\int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 + \mathrm { e } ^ { \pi } } { 2 }$$ Given that $$I _ { n } = \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x$$ show that, for \(n \geqslant 2\), $$I _ { n } = n ( n - 1 ) \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \cos ^ { 2 } x \sin ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$ and deduce that $$\left( n ^ { 2 } + 1 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 } .$$ A curve has equation \(y = \mathrm { e } ^ { x } \sin ^ { 5 } x\). Find, in an exact form, the mean value of \(y\) over the interval \(0 \leqslant x \leqslant \pi\).

8 The curve \(C\) has parametric equations \(x = \frac { 3 } { 2 } t ^ { 2 } , y = t ^ { 3 }\), for \(0 \leqslant t \leqslant 2\). Find the arc length of \(C\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 6\).

The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = ( 2 - t ) ^ { \frac { 1 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$ Find

1. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\),
2. the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 4\),
3. the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the \(x\)-axis and the \(y\)-axis.

9 The curve \(C\) has parametric equations $$x = 4 t + 2 t ^ { \frac { 3 } { 2 } } , \quad y = 4 t - 2 t ^ { \frac { 3 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 4$$ Find the arc length of \(C\), giving your answer correct to 3 significant figures. Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 32\).

9

1. Using the substitution \(u = \tan x\), or otherwise, find \(\int \sec ^ { 2 } x \tan ^ { 2 } x \mathrm {~d} x\).
   It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \tan ^ { 2 } x \mathrm {~d} x$$
2. Using the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \tan x \sec x\), show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = ( \sqrt { } 2 ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$
3. Hence find the mean value of \(\sec ^ { 4 } x \tan ^ { 2 } x\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\), giving your answer in exact form.