Volume with numerical methods

6 \includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-06_561_542_260_799} The diagram shows the curve \(y = \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).

1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region, giving your answer correct to 3 significant figures.
2. Find the exact volume of the solid formed when the shaded region is rotated completely about the \(x\)-axis.

6 \includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-10_351_488_264_826} The diagram shows the curve with equation \(y = \sqrt { } \left( 1 + 3 \cos ^ { 2 } \left( \frac { 1 } { 2 } x \right) \right)\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).

1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures.
2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced.

6 \includegraphics[max width=\textwidth, alt={}, center]{1b410c91-2fe9-46cf-8478-631b4165f98d-10_351_488_264_826} The diagram shows the curve with equation \(y = \sqrt { } \left( 1 + 3 \cos ^ { 2 } \left( \frac { 1 } { 2 } x \right) \right)\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).

1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures.
2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced.

7. The diagram shows the curve with equation \(y = 2 x - \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).

1. Find the area of the shaded region, giving your answer in terms of e. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Using Simpson's rule with two strips, estimate the volume of the solid formed.

6. \includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).

1. A measure of the effective voltage, \(M\) volts, in an electrical circuit is given by

$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$ where \(V\) volts is the voltage at time \(t\) seconds. Pairs of values of \(V\) and \(t\) are given in the following table. Use the trapezium rule with five values of \(V ^ { 2 }\) to estimate the value of \(M\). \begin{figure}[h] \end{figure} Figure 1 shows part of a curve \(C\) with equation \(y = x ^ { 2 } + 3\). The shaded region is bounded by \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 3\). The shaded region is rotated \(360 ^ { \circ }\) about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer as an exact multiple of \(\pi\).

7 The diagram shows part of the graph of \(y = \cos ^ { - 1 } x\) The diagram shows part of the graph of \(y = \cos ^ { - 1 } x\) \includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-07_689_958_358_539} The finite region enclosed by the graph of \(y = \cos ^ { - 1 } x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places.