Rotation about y-axis, standard curve

3 \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-04_700_401_260_870} The diagram shows part of the curve with equation \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the \(y\)-axis and the line \(y = 5\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the volume obtained.

5 \includegraphics[max width=\textwidth, alt={}, center]{1b5d8cb1-fd1b-4fcf-8975-b5d020991c9a-2_570_1050_1393_550} The diagram shows part of the curve \(x = \frac { 8 } { y ^ { 2 } } - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B ( 6,1 )\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

3 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_497_1106_554_515} The diagram shows part of the curve \(x = \frac { 12 } { y ^ { 2 } } - 2\). The shaded region is bounded by the curve, the \(y\)-axis and the lines \(y = 1\) and \(y = 2\). Showing all necessary working, find the volume, in terms of \(\pi\), when this shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.

1 \includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-2_668_554_260_797} The diagram shows part of the curve \(y = x ^ { 2 } + 1\). Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-18_650_611_260_762} The diagram shows a shaded region bounded by the \(y\)-axis, the line \(y = - 1\) and the part of the curve \(y = x ^ { 2 } + 4 x + 3\) for which \(x \geqslant - 2\).

1. Express \(y = x ^ { 2 } + 4 x + 3\) in the form \(y = ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. Hence, for \(x \geqslant - 2\), express \(x\) in terms of \(y\).
2. Hence, showing all necessary working, find the volume obtained when the shaded region 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.

7. (a) Find $$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$ giving your answer in its simplest form. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$ The curve \(C\) cuts the line \(y = 8\) at the point \(P\) with coordinates \(( k , 8 )\), where \(k\) is a constant.
(b) Find the value of \(k\). The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(y = 8\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(c) Find the exact value of the volume of the solid generated.

2. \includegraphics[max width=\textwidth, alt={}, center]{b124d427-1f9b-4770-95bb-ed79bae5b4fb-1_460_805_587_486} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \ln 3 x\).

1. Express the equation of the curve in the form \(x = \mathrm { f } ( y )\). The shaded region is bounded by the curve, the coordinate axes and the line \(y = 1\).
2. Find, in terms of \(\pi\) and e, the volume of the solid formed when the shaded region is rotated through four right angles about the \(y\)-axis.

3 Fig. 3 shows part of the curve \(y = 1 + x ^ { 2 }\), together with the line \(y = 2\). \begin{figure}[h] \end{figure} The region enclosed by the curve, the \(y\)-axis and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the volume of the solid generated, giving your answer in terms of \(\pi\).

3 Fig. 3 shows the curve \(y = \ln x\) and part of the line \(y = 2\). \begin{figure}[h] \end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.

1. Show that the volume of the solid of revolution formed is given by \(\int _ { 0 } ^ { 2 } \pi \mathrm { e } ^ { 2 y } \mathrm {~d} y\).
2. Evaluate this, leaving your answer in an exact form.

3 Fig. 3 shows the curve \(y = x ^ { 4 }\) and the line \(y = 4\). \begin{figure}[h] \end{figure} The finite region enclosed by the curve and the line is rotated through \(180 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated.

1 Fig. 6 shows the region enclosed by the curve \(y = \left( 1 + 2 x ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\) and the line \(y = 2\). \begin{figure}[h] \end{figure} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\).

4 The part of the curve \(y = 4 - x ^ { 2 }\) that is above the \(x\)-axis is rotated about the \(y\)-axis. This is shown in Fig. 4. Find the volume of revolution produced, giving your answer in terms of \(\pi\). \begin{figure}[h] \end{figure}

6 Fig. 3 shows the curve \(y = \ln x\) and part of the line \(y = 2\). \begin{figure}[h] \end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.

1. Show that the volume of the solid of revolution formed is given by \(\int _ { 0 } ^ { 2 } \pi \mathrm { e } ^ { 2 y } \mathrm {~d} y\).
2. Evaluate this, leaving your answer in an exact form.

2 Fig. 3 shows part of the curve \(y = 1 + x ^ { 2 }\), together with the line \(y = 2\). \begin{figure}[h] \end{figure} The region enclosed by the curve, the \(y\)-axis and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the volume of the solid generated, giving your answer in terms of \(\pi\).

9

1. Show that the derivative with respect to \(y\) of $$y \ln ( 2 y ) - y$$ is \(\ln ( 2 y )\).
2. \includegraphics[max width=\textwidth, alt={}, center]{390105da-0cba-4f82-8c8f-1f36090b1564-3_465_631_1859_717} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \mathrm { e } ^ { x ^ { 2 } }\). The point \(P \left( 2 , \frac { 1 } { 2 } \mathrm { e } ^ { 4 } \right)\) lies on the curve. The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = \frac { 1 } { 2 } e ^ { 4 }\). Find the exact volume of the solid produced when the shaded region is rotated completely about the \(y\)-axis.
3. Hence find the volume of the solid produced when the region bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\) is rotated completely about the \(y\)-axis. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

4 The part of the curve \(y = 4 - x ^ { 2 }\) that is above the \(x\)-axis is rotated about the \(y\)-axis. This is shown in Fig. 4. Find the volume of revolution produced, giving your answer in terms of \(\pi\). \begin{figure}[h] \end{figure}

6 Fig. 6 shows the region enclosed by the curve \(y = \left( 1 + 2 x ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\) and the line \(y = 2\). \begin{figure}[h] \end{figure} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\). \section*{Question 7 begins on page 4.}

6 The diagram shows the curve with equation \(y = \sqrt { 100 - 4 x ^ { 2 } }\), where \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}

1. Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis, giving your answer in terms of \(\pi\).
2. Use the mid-ordinate rule with five strips of equal width to find an estimate for \(\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The point \(P\) on the curve has coordinates \(( 3,8 )\).
   1. Find the gradient of the curve \(y = \sqrt { 100 - 4 x ^ { 2 } }\) at the point \(P\).
   2. Hence show that the equation of the tangent to the curve at the point \(P\) can be written as \(2 y + 3 x = 25\).
4. The shaded regions on the diagram below are bounded by the curve, the tangent at \(P\) and the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731} Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.

1. The region \(R\) is bounded by the curve \(x = \sin y\), the \(y\)-axis and the lines \(y = 1 , y = 3\). Find the volume of the solid generated when \(R\) is rotated through four right angles about the \(y\)-axis. Give your answer correct to two decimal places.
2. (a) Determine the number of solutions of the equations

$$\begin{array} { r } x + 2 y = 3 \\ 2 x - 5 y + 3 z = 8 \\ 6 y - 2 z = 0 \end{array}$$ (b) Give a geometric interpretation of your answer in part (a).

8. The curve \(y = 1 + x ^ { 3 }\) is denoted by \(C\).

1. A bowl is designed by rotating the arc of \(C\) joining the points \(( 0,1 )\) and \(( 2,9 )\) through four right angles about the \(y\)-axis. Calculate the capacity of the bowl.
2. Another bowl with capacity 25 is to be designed by rotating the arc of \(C\) joining the points with \(y\) coordinates 1 and \(a\) through four right angles about the \(y\)-axis. Calculate the value of \(a\).

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross-section, \(O A B C D E O\), of the design for a solid glass ornament. Figure 2 shows the finite region, \(R\), which is bounded by the \(y\)-axis, the horizontal line \(C B\), the vertical line \(B A\), and the curve \(A O\). The ornament is formed by rotating the region \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
The curve \(A O\) is modelled by the equation $$x = k y ^ { 2 } + \sqrt { y } \quad 0 \leqslant y \leqslant 4$$ where \(k\) is a constant.
The point \(A\) has coordinates ( \(0.4,4\) ) and the point \(B\) has coordinates ( \(0.4,4.5\) )
The units are centimetres.

1. Determine the value of \(k\) according to this model.
2. Use algebraic integration to determine the exact volume of glass that would be required to make the ornament, according to the model.
3. State a limitation of the model. When the ornament was manufactured, \(9 \mathrm {~cm} ^ { 3 }\) of glass was required.
4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.

7. \begin{figure}[h] \end{figure} A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student's design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = 1\), the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) through \(360 ^ { \circ }\) about the \(y\)-axis. The point \(A\) has coordinates ( \(1,0.5\) ) and the point \(B\) has coordinates ( \(0.5,2.5\) ) where the units are centimetres. The curve \(C _ { 1 }\) is modelled by the equation $$x = \frac { a } { y + b } \quad 0.5 \leqslant y \leqslant 2.5$$

1. Determine the value of \(a\) and the value of \(b\) according to the model. The curve \(C _ { 2 }\) is modelled to be an arc of the circle with centre \(( 0,3 )\).
2. Use calculus to determine the volume of plastic required to make the chess piece according to the model.

12 The shaded region shown in the diagram below is bounded by the \(x\)-axis, the curve \(y = \mathrm { f } ( x )\), and the lines \(x = a\) and \(x = b\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-16_661_721_406_662} The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
12

1. Show that the volume of this solid is $$\pi \int _ { a } ^ { b } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x$$ 12
2. In the case where \(a = 1 , b = 2\) and $$f ( x ) = \frac { x + 3 } { ( x + 1 ) \sqrt { x } }$$ show that the volume of the solid is $$\pi \left( \ln \left( \frac { 2 ^ { m } } { 3 ^ { n } } \right) - \frac { 2 } { 3 } \right)$$ where \(m\) and \(n\) are integers.

16 The function f is defined by $$f ( x ) = \frac { a x + 5 } { x + b }$$ where \(a\) and \(b\) are constants. The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = - 2\) and \(y = 3\) 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The diagram shows the graph of \(y = \mathrm { f } ( x )\) and its asymptotes.
   The shaded region \(R\) is enclosed by the graph of \(y = \mathrm { f } ( x )\), the \(x\)-axis and the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-20_858_1002_1267_504} 16
3. 1. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. 16
4. (ii) The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis to form a solid.
   Find the volume of this solid.
   Give your answer to three significant figures.
   [0pt] [4 marks]