Centre of mass of solid of revolution

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the region \(R\) bounded by the curve with equation \(y = \mathrm { e } ^ { - x }\), the line \(x = 1\), the \(x\)-axis and the \(y\)-axis. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac { \pi } { 2 } \left( 1 - \mathrm { e } ^ { - 2 } \right)\).
2. Find, in terms of e, the distance of the centre of mass of \(S\) from \(O\).

1. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y ^ { 2 } = 9 ( 4 - x )\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 1. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(S\).

5. \begin{figure}[h] \end{figure} Figure 3 shows the finite region \(R\) which is bounded by part of the curve with equation \(y = \sin x\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\). A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { \pi ^ { 2 } } { 4 }\)
2. find, in terms of \(\pi\), the \(x\) coordinate of the centre of mass of \(S\).

1. \begin{figure}[h] \end{figure} The region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \frac { 1 } { x }\), the line with equation \(x = 1\), the positive \(x\)-axis and the line with equation \(x = a\) where \(a > 1\) A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the volume of \(S\) is $$\pi \left( 1 - \frac { 1 } { a } \right)$$
2. Find the \(x\) coordinate of the centre of mass of \(S\).

1. \begin{figure}[h] \end{figure} The shaded region R is bounded by the x -axis, the line with equation \(\mathrm { x } = 1\), the curve with equation \(y = 1 + \sqrt { x }\) and the y-axis, as shown in Figure 1. The unit of length on both of the axes is 1 m . The region R is rotated through \(2 \pi\) radians about the x-axis to form a solid of revolution which is used to model a uniform solid \(S\). Show, using the model and algebraic integration, that

1. the volume of \(S\) is \(\frac { 17 \pi } { 6 } \mathrm {~m} ^ { 3 }\)
2. the centre of mass of \(S\) is \(\frac { 49 } { 85 } \mathrm {~m}\) from 0 . \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_2264_41_314_1987}

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
   The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
4. Find the size of the angle between \(V A\) and the vertical.

1. The region enclosed by the curve with equation \(y = \frac { 1 } { 2 } \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 4\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\). Use algebraic integration to find the exact value of the \(x\) coordinate of the centre of mass of \(S\).
   (6)

"都 D \(\_\_\_\_\) 1

1. The finite region enclosed by the curve with equation \(y = 3 - \sqrt { x }\) and the lines \(x = 0\) and \(y = 0\) is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a uniform solid \(S\).

Use algebraic integration to

1. show that the volume of \(S\) is \(\frac { 27 } { 2 } \pi\)
2. find the \(x\) coordinate of the centre of mass of \(S\).

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_483_469_1402_767} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
   The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
4. Find the size of the angle between \(V A\) and the vertical.

   Leave
   blank
   Q6

   VIIIV SIHI NI JAIIM ION OC

   VIIIV SIHI NI JIHMM ION OO

   VI4V SIHI NI JIIYM IONOO

2. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = 9 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 1. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of \(S\).

5. \begin{figure}[h] \end{figure} Figure 4 shows the region \(R\) bounded by part of the curve with equation \(y = \cos x\), the \(x\)-axis and the \(y\)-axis. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac { \pi ^ { 2 } } { 4 }\)
2. Find, using algebraic integration, the \(x\) coordinate of the centre of mass of \(S\).

2. The finite region bounded by the \(x\)-axis, the curve with equation \(y = 2 \mathrm { e } ^ { x }\), the \(y\)-axis and the line \(x = 1\) is rotated through one complete revolution about the \(x\)-axis to form a uniform solid. Use algebraic integration to

1. show that the volume of the solid is \(2 \pi \left( \mathrm { e } ^ { 2 } - 1 \right)\),
2. find, in terms of e, the \(x\) coordinate of the centre of mass of the solid.

1 The region bounded by the \(x\)-axis, the curve \(\mathrm { y } = \sqrt { 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x - 20 }\) and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution \(R\). The centre of mass of \(R\) is the point \(G\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-2_569_463_507_280}

1. Explain why the \(y\)-coordinate of \(G\) is 0 .
2. Find the \(x\)-coordinate of \(G\). \(P\) is a point on the edge of the curved surface of \(R\) where \(x = 4 . R\) is freely suspended from \(P\) and hangs in equilibrium.
3. Find the angle between the axis of symmetry of \(R\) and the vertical.

6. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
   (3 marks)

3 The region between the curve \(y = x \sqrt { } ( 3 - x )\) and the \(x\)-axis for \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.

2 The region \(R\) is bounded by the curve \(y = \sqrt { 4 a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), the \(x\)-axis, the \(y\)-axis and the line \(x = a\), where \(a\) is a positive constant. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.

2 The region \(R\) is bounded by the \(x\)-axis, the lines \(x = a\) and \(x = 2 a\), and the curve \(y = \frac { a ^ { 3 } } { x ^ { 2 } }\) for \(a \leqslant x \leqslant 2 a\), where \(a\) is a positive constant. A uniform solid of revolution is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.

2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis and the curve \(y = x \left( 1 - \frac { x ^ { 2 } } { a ^ { 2 } } \right)\) for \(0 \leqslant x \leqslant a\), where \(a\) is a constant, about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.

\includegraphics{figure_2} The region \(R\) is bounded by the curve with equation \(y = e^x\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2\pi\) about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac{1}{2}\pi (e^4 - e^2)\). [4]
2. Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\). [6]

A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt{x}\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\). [5]

The finite region bounded by the \(x\)-axis, the curve \(y = \frac{1}{x}\), the line \(x = \frac{1}{4}\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.

1. Show that the volume of the solid is \(21\pi\). [4]
2. Find the coordinates of the centre of mass of the solid. [5]

In this question you must show detailed reasoning. \includegraphics{figure_7} Fig. 7 shows the curve with equation \(y = \frac{2}{3}\ln x\). The region R, shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(x\)-coordinate of the centre of mass of the solid. [8]

[In this question you may use the facts that for a uniform solid right circular cone of height \(h\) and base radius \(r\) the volume is \(\frac{1}{3}\pi r^2 h\) and the centre of mass is \(\frac{1}{4}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_9} The diagram shows the shaded region S bounded by the curve \(y = e^{\frac{1}{4}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis, and the line \(y = \frac{1}{4}e(6-x)\). The line \(y = \frac{1}{4}e(6-x)\) meets the curve \(y = e^{\frac{1}{4}x}\) at the point A with coordinates \((2, e)\). The region S is rotated through \(2\pi\) radians about the \(x\)-axis to form a uniform solid of revolution T.

1. Show that the \(x\)-coordinate of the centre of mass of T is \(\frac{3(5e^2 + 1)}{7e^2 - 3}\). [8]

Solid T is freely suspended from A and hangs in equilibrium.

1. Determine the angle between AO, where O is the origin, and the vertical. [3]