Centre of mass of lamina by integration

2. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of the region \(R\) which is bounded by the curve with equation \(y = \frac { 3 } { x ^ { 2 } }\), the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis, as shown in Figure 1. The centre of mass of the lamina has coordinates \(( \bar { x } , \bar { y } )\).
Use algebraic integration to find

1. the value of \(\bar { x }\),
2. the value of \(\bar { y }\).

1. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of the region \(R\).
Region \(R\) is bounded by the curve with equation \(y = x ( x + a )\) where \(a\) is a positive constant, the positive \(x\)-axis and the line with equation \(x = a\), as shown shaded in Figure 1. Find the \(\boldsymbol { y }\) coordinate of the centre of mass of the lamina.

2. \begin{figure}[h] \end{figure} A uniform lamina \(L\) is in the shape of an equilateral triangle of side \(2 a\). The lamina is placed in the \(x y\)-plane with one vertex at the origin \(O\) and an axis of symmetry along the \(x\)-axis, as shown in Figure 1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(L\).

1. In this question you must show all stages in your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = 3\), the curve with equation \(y = \sqrt { ( x + 1 ) }\) and the \(y\)-axis.
Find the \(\boldsymbol { y }\) coordinate of the centre of mass of a uniform lamina in the shape of \(R\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) in which \(A B = 6 \mathrm {~cm} , B C = 9 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The centre of mass of the lamina is \(G\). Use algebraic integration to find the distance of \(G\) from \(A B\).
(6)

1. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of the region \(R\). Region \(R\) is bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown shaded in Figure 1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of the lamina.

9 The curve \(C\) has equation \(y = x ^ { \frac { 3 } { 2 } }\). Find the coordinates of the centroid of the region bounded by \(C\), the lines \(x = 1 , x = 4\) and the \(x\)-axis. Show that the length of the arc of \(C\) from the point where \(x = 5\) to the point where \(x = 28\) is 139 .

10 The curve \(C\) has equation $$y = 2 \left( \frac { x } { 3 } \right) ^ { \frac { 3 } { 2 } }$$ where \(0 \leqslant x \leqslant 3\). Show that the arc length of \(C\) is \(2 ( 2 \sqrt { 2 } - 1 )\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 3\).

8 The curve \(C\) has parametric equations \(x = \frac { 3 } { 2 } t ^ { 2 } , y = t ^ { 3 }\), for \(0 \leqslant t \leqslant 2\). Find the arc length of \(C\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 6\).

The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant 4\).

1. The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = 4\). Find, in terms of e, the coordinates of the centroid of the region \(R\).
2. Show that \(\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\), where \(s\) denotes the arc length of \(C\), and find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

3 A finite region \(R\) in the \(x - y\) plane is bounded by the curve with equation \(y = \sqrt { } x - \frac { 1 } { \sqrt { } x }\), the \(x\)-axis between \(x = 1\) and \(x = 4\), and the line \(x = 4\). Find the exact value of the \(y\)-coordinate of the centroid of \(R\).

2 The curve \(C\) has equation \(y = 2 x ^ { \frac { 1 } { 2 } }\) for \(0 \leqslant x \leqslant 4\). Find

1. the mean value of \(y\) with respect to \(x\) for \(0 \leqslant x \leqslant 4\),
2. the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x = 4\) and the \(x\)-axis.

7 The curve \(C\) has equation \(y = \mathrm { e } ^ { - 2 x }\). Find, giving your answers correct to 3 significant figures,

1. the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) over the interval \(0 \leqslant x \leqslant 2\),
2. the coordinates of the centroid of the region bounded by \(C\), \(x = 0\), \(x = 2\) and \(y = 0\).

1 The curve \(C\) has equation \(y = x ^ { a }\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis.

4 Fig. 4.1 shows the region \(R\) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 8\). \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of the centre of mass of a uniform solid of revolution obtained by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
2. Find the coordinates of the centre of mass of a uniform lamina in the shape of the region \(R\).
3. Using your answer to part (ii), or otherwise, find the coordinates of the centre of mass of a uniform lamina in the shape of the region (shown shaded in Fig. 4.2) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the line \(y = \frac { 1 } { 2 }\) and the line \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-4_595_1015_1610_607} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_705_501_260_863} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The region \(R\), shown in Fig. 4.1, is bounded by the curve \(x ^ { 2 } - y ^ { 2 } = k ^ { 2 }\) for \(k \leqslant x \leqslant 4 k\) and the line \(x = 4 k\), where \(k\) is a positive constant. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when \(R\) is rotated about the \(x\)-axis.
2. A uniform lamina occupies the region bounded by the curve \(y = \frac { x ^ { 3 } } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant 2 a\), the \(x\)-axis and the line \(x = 2 a\), where \(a\) is a positive constant. The vertices of the lamina are \(\mathrm { O } ( 0,0 ) , \mathrm { A } ( 2 a , 8 a )\) and \(\mathrm { B } ( 2 a , 0 )\), as shown in Fig. 4.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_714_509_1546_858} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Find the coordinates of the centre of mass of the lamina.
   2. The lamina is freely suspended from the point A and hangs in equilibrium. Find the angle that AB makes with the vertical.

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 3\), and the line \(x = \ln 3\). Find, in an exact form, the coordinates of the centre of mass of this lamina.
2. A region is bounded by the \(x\)-axis, the curve \(y = \frac { 6 } { x ^ { 2 } }\) for \(2 \leqslant x \leqslant a\) (where \(a > 2\) ), the line \(x = 2\) and the line \(x = a\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
   1. Show that the \(x\)-coordinate of the centre of mass of this solid is \(\frac { 3 \left( a ^ { 3 } - 4 a \right) } { a ^ { 3 } - 8 }\).
   2. Show that, however large the value of \(a\), the centre of mass of this solid is less than 3 units from the origin.

3 In this question, give your answers in an exact form.
The region \(R _ { 1 }\) (shown in Fig. 3) is bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 5\), and the curve \(y = \frac { 1 } { x }\) for \(1 \leqslant x \leqslant 5\).

1. A uniform solid of revolution is formed by rotating the region \(R _ { 1 }\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
2. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-4_849_841_735_651} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The region \(R _ { 2 }\) is bounded by the \(y\)-axis, the lines \(y = 1\) and \(y = 5\), and the curve \(y = \frac { 1 } { x }\) for \(\frac { 1 } { 5 } \leqslant x \leqslant 1\). The region \(R _ { 3 }\) is the square with vertices \(( 0,0 ) , ( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\).
3. Write down the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 2 }\).
4. Find the coordinates of the centre of mass of a uniform lamina occupying the region consisting of \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) (shown shaded in Fig. 3).

4 The region \(A\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, the \(y\)-axis and the line \(x = 3\). The region \(B\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(y\)-axis and the line \(y = 14\). These regions are shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(A\).
2. The region \(B\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(R\). Find the \(y\)-coordinate of the centre of mass of the solid \(R\).
3. The region \(A\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(S\). Using your answer to part (ii), or otherwise, find the \(y\)-coordinate of the centre of mass of the solid \(S\).

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis and the curve \(y = \frac { x ^ { 2 } ( a - x ) } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant a\). Find the coordinates of the centre of mass of this lamina.
2. The region \(A\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(x = 3\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(y = 5\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-5_604_460_605_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(A\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   2. Using your answer to part (i), or otherwise, find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. \section*{END OF QUESTION PAPER}

5 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 5\) and the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 5\), is occupied by a uniform lamina.

1. Show that the centre of mass of this lamina has \(x\)-coordinate $$\frac { 5 } { 4 } \ln 5 - 1$$
2. Find the \(y\)-coordinate of the centre of mass.

2 The region enclosed by the curve \(y = \sqrt { } x\) for \(0 \leqslant x \leqslant 9\), the \(x\)-axis and the line \(x = 9\) is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.

3 The region bounded by the curve \(y = 2 x + x ^ { 2 }\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, and the line \(x = 3\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.

2 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 3\), and the curve \(y = \mathrm { e } ^ { - x }\) for \(0 \leqslant x \leqslant \ln 3\), is occupied by a uniform lamina. Find, in an exact form, the coordinates of the centre of mass of this lamina.

3 The region bounded by the \(y\)-axis and the curves \(y = \sin 2 x\) and \(y = \sqrt { 2 } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) is occupied by a uniform lamina. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.