Rotation about x-axis, standard curve

2 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_633_787_402_680} The diagram shows the curve \(y = 3 x ^ { \frac { 1 } { 4 } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\). Find the volume of the solid obtained when this shaded region is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).

9 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_391_595_1978_774} The diagram shows part of the curve \(y = \frac { 6 } { 3 x - 2 }\).

1. Find the gradient of the curve at the point where \(x = 2\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).

2 \includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-2_486_727_625_708} The diagram shows part of the curve \(y = \frac { a } { x }\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(24 \pi\), find the value of \(a\).

3

1. Sketch the curve \(y = ( x - 2 ) ^ { 2 }\).
2. The region enclosed by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained, giving your answer in terms of \(\pi\).

1 \includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-2_618_533_246_808} The diagram shows the region enclosed by the curve \(y = \frac { 6 } { 2 x - 3 }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

2 \includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-2_627_551_429_790} The diagram shows part of the curve \(y = \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }\) and the point \(P ( 2,3 )\) lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

8 The function f is such that \(\mathrm { f } ( x ) = \frac { 3 } { 2 x + 5 }\) for \(x \in \mathbb { R } , x \neq - 2.5\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and explain why f is a decreasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. A curve has the equation \(y = \mathrm { f } ( x )\). Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 The equation of a curve is \(y = \frac { 9 } { 2 - x }\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine, with a reason, whether the curve has any stationary points.
2. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points. \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. University of 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. }

3. \begin{figure}[h] \end{figure} The curve shown in Figure 2 has equation \(y = \frac { 1 } { ( 2 x + 1 ) }\). The finite region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = b\) is shown shaded in Figure 2. This region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to generate a solid of revolution. Find the volume of the solid generated. Express your answer as a single simplified fraction, in terms of \(a\) and \(b\).

2. \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-06_974_1088_116_548} \section*{Figure 1} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { 9 } { ( 2 x - 3 ) ^ { 1.25 } } \quad x > \frac { 3 } { 2 }$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(y = 9\) and the line with equation \(x = 6\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution. Find, by algebraic integration, the exact volume of the solid generated.

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve \(y = \frac { 3 } { \sqrt { } ( 1 + 4 x ) }\). The region \(R\) is bounded by the curve, the \(x\)-axis, and the lines \(x = 0\) and \(x = 2\), as shown shaded in Figure 1.

1. Use integration to find the area of \(R\). The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
2. Use integration to find the exact value of the volume of the solid formed.

5. The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 5\).

1. Find the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Find the volume of the solid formed, giving your answer in the form \(k \pi \ln 2\).

1. The region bounded by the curve \(y = x ^ { 2 } - 2 x\) and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

Find the volume of the solid formed, giving your answer in terms of \(\pi\).

4. The finite region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

1. Find the exact area of \(R\).
2. Show that the volume of the solid formed when \(R\) is rotated through four right angles about the \(x\)-axis is \(\frac { 1 } { 3 } \pi\).

2. The diagram shows the curve with equation \(y = x \sqrt { 2 - x } , 0 \leq x \leq 2\).
Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

7. The finite region \(R\) is bounded by the curve with equation \(y = x + \frac { 2 } { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).

1. Find the exact area of \(R\). The region \(R\) is rotated completely about the \(x\)-axis.
2. Find the volume of the solid formed, giving your answer in terms of \(\pi\).

5. The finite region \(R\) is bounded by the curve with equation \(y = \sqrt [ 3 ] { 3 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 2 } { 3 }\) and \(x = 3\).

1. Find the area of \(R\).
2. Find, in terms of \(\pi\), the volume of the solid formed when \(R\) is rotated through four right angles about the \(x\)-axis.

6 \includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-3_483_956_264_593} The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

1. Find the exact area of the shaded region.
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer.

5

1. Find \(\int ( 3 x + 7 ) ^ { 9 } \mathrm {~d} x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_537_881_402_671} The diagram shows the curve \(y = \frac { 1 } { 2 \sqrt { x } }\). The shaded region is bounded by the curve and the lines \(x = 3 , x = 6\) and \(y = 0\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced, simplifying your answer.

4

1. \includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-2_579_785_1279_721} The diagram shows the curve \(y = \frac { 2 } { \sqrt { } x }\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1 , x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed.
2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int _ { 1 } ^ { 5 } \sqrt { } \left( x ^ { 2 } + 1 \right) \mathrm { d } x ,$$ giving your answer correct to 3 decimal places.

4 \includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-2_419_707_1576_660} The diagram shows the curve $$y = \frac { 1 } { \sqrt { } ( 4 x + 1 ) }$$ The region \(R\) (shaded in the diagram) is enclosed by the curve, the axes and the line \(x = 2\).

1. Show that the exact area of \(R\) is 1 .
2. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed.

2 The graph shows part of the curve \(y = x ^ { 2 } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{62dbc58e-f498-483f-a9aa-05cb5aa44881-2_380_876_715_575} Find the volume when the area between this curve, the axes and the line \(x = 2\) is rotated through \(360 ^ { 0 }\) about the \(x\)-axis.

3 The curve \(y ^ { 2 } = x - 1\) for \(1 \leq x \leq 3\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed.

3 The graph shows part of the curve \(y ^ { 2 } = ( x - 1 )\). \includegraphics[max width=\textwidth, alt={}, center]{73112db3-7b05-48db-9fff-fdbac7dbd564-2_428_860_973_616} Find the volume when the area between this curve, the \(x\)-axis and the line \(x = 5\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

5 \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_559_1191_1749_479} The diagram shows the curve with equation \(y = \frac { 6 } { \sqrt { 3 x - 2 } }\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 1 , x = a\) and \(y = 0\), where \(a\) is a constant greater than 1 . It is given that the area of \(R\) is 16 square units. Find the value of \(a\) and hence find the exact volume of the solid formed when \(R\) is rotated completely about the \(x\)-axis.
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