1.08d Evaluate definite integrals: between limits

4. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is in the shape of the region \(R\).
Region \(R\) lies in the first quadrant and is bounded by the curve with equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1\), the \(x\)-axis, and the \(y\)-axis, as shown shaded in Figure 3. The point \(A\) is the point of intersection of the curve and the \(x\)-axis.
The point \(B\) is the point of intersection of the curve and the \(y\)-axis.
One unit on each axis represents 1 m .
The area of \(R\) is \(6 \pi\) The centre of mass of \(R\) lies at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. Use algebraic integration to show that \(\bar { x } = \frac { 16 } { 3 \pi }\)
2. Use algebraic integration to find the exact value of \(\bar { y }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(O A\) at angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)

1. A flag pole is 15 m long.

The flag pole is non-uniform so that, at a distance \(x\) metres from its base, the mass per unit length of the flag pole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\) is given by the formula \(m = 10 \left( 1 - \frac { x } { 25 } \right)\). The flag pole is modelled as a rod.

1. Show that the mass of the flag pole is 105 kg .
2. Find the distance of the centre of mass of the flag pole from its base.

8 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }\). The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 8\) and \(x = a\), where \(a > 8\). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of \(a\).

8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).

10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$

1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]

   \(n\)	0	1	2	3	4	5	6	7
   \(\mathrm { x } _ { \mathrm { n } }\)	1	0.958509	0.950084	0.948261	0.94786	0.947772	0.947753	0.947748

   \captionsetup{labelformat=empty} \caption{Fig. 10.1}
   \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]

   \(x\)	\(y\)
   0.9477475	\(- 7.79967 \mathrm { E } - 07\)
   0.9477485	\(- 2.90821 \mathrm { E } - 06\)
   \(x\)	\(y\)
   0.947745	\(4.54066 \mathrm { E } - 06\)
   0.947755	\(- 1.67417 \mathrm { E } - 05\)

   \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{table}
3. Consider the information in Fig. 10.1 and Fig. 10.2.

14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h] \end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.

3 In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = x ^ { 5 }\) and the square \(O A B C\) where the points \(A , B\) and \(C\) have coordinates \(( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\) respectively. The curve cuts the square into two parts. \includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-04_658_780_1318_230} Show that the relationship between the areas of the two parts of the square is \(\frac { \text { Area to left of curve } } { \text { Area below curve } } = 5\).

5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h] \end{figure} Find the area of the part of the circle below the \(x\)-axis.

1 In this question you must show detailed reasoning. Show that \(\int _ { 4 } ^ { 9 } ( 2 x + \sqrt { x } ) \mathrm { d } x = \frac { 233 } { 3 }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-6_533_524_246_772} The diagram shows a container which consists of a cylinder with a solid base and a hemispherical top. The radius of the cylinder is \(r \mathrm {~cm}\) and the height is \(h \mathrm {~cm}\). The container is to be made of thin plastic. The volume of the container is \(45 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the container, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { 5 } { 3 } \pi r ^ { 2 } + \frac { 90 \pi } { r } .$$ [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and the surface area of a sphere is \(S = 4 \pi r ^ { 2 }\).]
2. Use calculus to find the minimum surface area of the container, justifying that it is a minimum.
3. Suggest a reason why the manufacturer would wish to minimise the surface area.

1

1. Simplify:
   1. \(x ^ { \frac { 3 } { 2 } } \times x ^ { \frac { 1 } { 2 } }\);
   2. \(x ^ { \frac { 3 } { 2 } } \div x\);
   3. \(\left( x ^ { \frac { 3 } { 2 } } \right) ^ { 2 }\).
2. 1. Find \(\int 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Hence find the value of \(\int _ { 1 } ^ { 9 } 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).

5. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$

1. Find \(\mathrm { f } ( x )\).
2. Show that the area of the finite region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is \(4 \frac { 1 } { 2 }\).

8

1. Write down \(\int \sec ^ { 2 } x \mathrm {~d} x\).
2. Given that \(y = \frac { \cos x } { \sin x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
3. Prove the identity \(( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x\).
4. Hence find \(\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x\), giving your answer to two significant figures.

6

1. Find the first three non-zero terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }\) for \(| x | < 2\).
2. Use this result to find an approximation for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to
   4 significant figures.
3. Given that \(\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c\), evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to 4 significant figures.