Rotation about y-axis, region between two curves

A question is this type if and only if it asks for the volume when a region bounded by two distinct curves (or a curve and a non-axis line) is rotated about the y-axis, requiring subtraction of volumes or use of cylindrical shells.

4 questions · Challenging +1.1

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CAIE P1 2023 March Q11
11 marks Challenging +1.2
11 \includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-16_599_780_274_671} The diagram shows the curve with equation \(x = y ^ { 2 } + 1\). The points \(A ( 5,2 )\) and \(B ( 2 , - 1 )\) lie on the curve.
  1. Find an equation of the line \(A B\).
  2. Find the volume of revolution when the region between the curve and the line \(A B\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-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.
CAIE P1 2011 November Q10
Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-4_764_929_255_609} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt { } ( x + 1 )\), meeting at \(( - 1,0 )\) and \(( 0,1 )\).
  1. Find the area of the shaded region.
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
OCR MEI C4 Q3
7 marks Challenging +1.2
3 Fig. 6 shows the region enclosed by part of the curve \(y = 2 x ^ { 2 }\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(\mathrm { P } ( 1,2 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d786d33-c5c2-44a6-8273-7a3e43e552ef-3_640_923_399_613} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed.
[0pt] [You may use the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]
OCR MEI C4 2011 June Q6
7 marks Challenging +1.2
6 Fig. 6 shows the region enclosed by part of the curve \(y = 2 x ^ { 2 }\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(\mathrm { P } ( 1,2 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0a6247c9-ba64-4a8f-9e10-83986136cf56-2_643_933_1667_607} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed.
[0pt] [You may use the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] Section B (36 marks)