Rotation about x-axis, region between two curves

9 \includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-10_540_1113_260_516} The diagram shows part of the curve with equation \(y ^ { 2 } = x - 2\) and the lines \(x = 5\) and \(y = 1\). The shaded region enclosed by the curve and the lines is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained.

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The diagram shows the straight line \(x + y = 5\) intersecting the curve \(y = \frac { 4 } { x }\) at the points \(A ( 1,4 )\) and \(B ( 4,1 )\). Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-5_609_897_255_625} The diagram shows part of the curve \(y = \frac { 1 } { ( 3 x + 1 ) ^ { \frac { 1 } { 4 } } }\). The curve cuts the \(y\)-axis at \(A\) and the line \(x = 5\) at \(B\).

1. Show that the equation of the line \(A B\) is \(y = - \frac { 1 } { 10 } x + 1\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

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.

4. \includegraphics[max width=\textwidth, alt={}, center]{c0b79c3c-9537-4c71-903b-01434dfb5d26-1_492_803_1562_452} The diagram shows the curves \(y = ( x - 1 ) ^ { 2 }\) and \(y = 2 - \frac { 2 } { x } , x > 0\).

1. Verify that the two curves meet at the points where \(x = 1\) and where \(x = 2\). The shaded region bounded by the two curves is rotated completely about the \(x\)-axis.
2. Find the exact volume of the solid formed.

6 \includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-3_586_798_267_676} The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.

6 \includegraphics[max width=\textwidth, alt={}, center]{cf154c94-6248-4dda-91e8-61349cc10482-3_606_846_251_614} The diagram shows the curves \(y = \cos x\) and \(y = \sin x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The region \(R\) is bounded by the curves and the \(x\)-axis. Find the volume of the solid of revolution formed when \(R\) is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).

\includegraphics{figure_4} The diagram shows the curves \(y = e^{3x}\) and \(y = (2x - 1)^4\). The shaded region is bounded by the two curves and the line \(x = \frac{1}{2}\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [7]