Multi-part: volume and stationary points

9 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-4_602_899_248_625} The diagram shows part of the curve \(y = x + \frac { 4 } { x }\) which has a minimum point at \(M\). The line \(y = 5\) intersects the curve at the points \(A\) and \(B\).

1. Find the coordinates of \(A , B\) and \(M\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{5e2a6be8-b260-4b03-90c5-7485adbc39cc-3_513_1023_1838_561} The diagram shows part of the curve \(y = 4 \sqrt { } x - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).

1. Find the coordinates of \(A\) and \(M\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).

10 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826} The diagram shows the part of the curve \(y = \frac { 8 } { x } + 2 x\) for \(x > 0\), and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y ^ { 2 } \mathrm {~d} x\).
2. Find the coordinates of \(M\) and determine the coordinates and nature of the stationary point on the part of the curve for which \(x < 0\).
3. Find the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 A curve has equation \(y = x ^ { 2 } + \frac { 2 } { x }\).

1. Write down expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine its nature.
3. Find the volume of the solid formed when the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated completely about the \(x\)-axis.

8 The equation of a curve is \(y = \sqrt { } \left( 8 x - x ^ { 2 } \right)\). Find

1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and the coordinates of the stationary point on the curve,
2. the volume obtained when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 A curve has equation \(y = ( k x - 3 ) ^ { - 1 } + ( k x - 3 )\), where \(k\) is a non-zero constant.

1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point, justifying your answers.
2. \includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-4_556_855_1032_685} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

9 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-4_547_1401_264_370} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \sqrt { } ( 1 + 2 x )\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).

1. Find the \(x\)-coordinate of \(M\).
2. Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

11 \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5. \begin{figure}[h] \end{figure} Figure 1 shows a graph of \(y = x \sqrt { } \sin x , 0 < x < \pi\). The maximum point on the curve is \(A\).

1. Show that the \(x\)-coordinate of the point \(A\) satisfies the equation \(2 \tan x + x = 0\). The finite region enclosed by the curve and the \(x\)-axis is shaded as shown in Fig. 1.
   A solid body \(S\) is generated by rotating this region through \(2 \pi\) radians about the \(x\)-axis.
2. Find the exact value of the volume of \(S\).
   (7)