Rotation about y-axis, region between two curves

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.

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] \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.]

10 \includegraphics[max width=\textwidth, alt={}, center]{a3cad2ad-e06b-4aa4-a3a9-a2840cd54893-3_529_527_264_810} The diagram shows the region \(R\) in the first quadrant bounded by the curves \(y = \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right)\) and \(y = \frac { 1 } { 5 } \left( 9 - x ^ { 2 } \right)\). \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Calculate the volume of the solid formed.

\includegraphics{figure_4} The diagram shows part of the curve \(y = \frac{k}{x}\), where \(k\) is a positive constant. The points A and B on the curve have \(x\)-coordinates 2 and 6 respectively. Lines through A and B parallel to the axes as shown meet at the point C. The region R is bounded by the curve and the lines \(x = 2\), \(x = 6\) and \(y = 0\). The region S is bounded by the curve and the lines AC and BC. It is given that the area of the region R is \(\ln 81\).

1. Show that \(k = 4\). [3]
2. Find the exact volume of the solid produced when the region S is rotated completely about the \(x\)-axis. [4]

Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at P (1, 2). \includegraphics{figure_6} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7] [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.]

Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(P(1, 2)\). \includegraphics{figure_1} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7]