Rotation about x-axis: rational or reciprocal function

A question asking for the volume when a region bounded by a rational or reciprocal function (e.g., a/x, 1/(ax+b), 6/sqrt(3x+1)) and lines parallel to the axes is rotated about the x-axis.

16 questions · Moderate -0.0

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CAIE P1 2009 June Q9
8 marks Moderate -0.3
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 \(\pi\).
CAIE P1 2009 November Q8
9 marks Moderate -0.3
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.
CAIE P1 2010 November Q11
11 marks Standard +0.3
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.
Edexcel C4 2008 January Q3
5 marks Standard +0.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ac7d862f-d10d-45ed-9077-ae4c7413cbf6-04_493_490_278_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\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 \(b\).
OCR C3 2008 January Q5
8 marks Moderate -0.3
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.
OCR C3 2011 January Q5
9 marks Standard +0.3
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.
[0pt] [9]
OCR C3 2012 January Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-2_490_713_447_660} The diagram shows part of the curve \(y = \frac { 6 } { ( 2 x + 1 ) ^ { 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 1\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis.
Edexcel PURE 2024 October Q7
Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-18_510_680_251_696} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = \frac { 3 x - 1 } { x + 2 } \quad x > - 2$$
  1. Show that $$\frac { 3 x - 1 } { x + 2 } \equiv A + \frac { B } { x + 2 }$$ where \(A\) and \(B\) are constants to be found. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 1\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Use the answer to part (a) and algebraic integration to find the exact volume of the solid generated, giving your answer in the form $$\pi ( p + q \ln 2 )$$ where \(p\) and \(q\) are rational constants.
CAIE P1 2024 June Q9
6 marks Standard +0.3
\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \frac{1}{(5x - 4)^3}\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \((1, 1)\). Find the exact volume of the solid generated when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]
CAIE P1 2010 June Q2
4 marks Moderate -0.3
\includegraphics{figure_2} 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 \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]
CAIE P1 2012 June Q1
4 marks Moderate -0.3
\includegraphics{figure_1} The diagram shows the region enclosed by the curve \(y = \frac{6}{2x - 3}\), 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 \(360°\) about the \(x\)-axis. [4]
Edexcel C4 Q10
8 marks Standard +0.3
\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]
Edexcel C4 Q26
8 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = 1 + \frac{1}{2\sqrt{x}}\). The shaded region \(R\), bounded by the curve, that \(x\)-axis and the lines \(x = 1\) and \(x = 4\), is rotated through \(360°\) about the \(x\)-axis. Using integration, show that the volume of the solid generated is \(\pi (5 + \frac{1}{2} \ln 2)\). [8]
OCR C3 Q4
8 marks Standard +0.2
  1. \includegraphics{figure_4a} 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. [4]
  2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int_1^5 \sqrt{(x^2 + 1)} \, dx,$$ giving your answer correct to 3 decimal places. [4]
OCR C3 Q5
8 marks Moderate -0.8
  1. Find \(\int (3x + 7)^9 \, dx\). [3]
  2. \includegraphics{figure_5b} 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. [5]
Edexcel C4 Q2
8 marks Standard +0.3
\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]