Centre of mass by integrationAQA A-Level Further Maths: Subtopic test

1. 1
   A uniform lamina occupies the region $R$ bounded by the curve $y=x^2$, the $x$-axis and the line $x=2$. Lengths are in metres.

   (a)

   Find the $x$-coordinate of the centre of mass of the lamina.

   [1 mark]

   • A$1.0$ m
   • B$1.5$ m
   • C$4.0$ m
   • D$\frac43$ m

   (b)

   Find the $y$-coordinate of the centre of mass of the lamina.

   [1 mark]

   • A$1.2$ m
   • B$2.4$ m
   • C$3.2$ m
   • D$1.3$ m

   (c)

   A second uniform lamina occupies the region bounded by $y=x^2$, the $x$-axis and the line $x=3$. Find its $x$-coordinate of the centre of mass.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The region $R$ is bounded by the curve $y=\sqrt{x}$, the $x$-axis and the line $x=4$. A uniform solid is formed by rotating $R$ through $2\pi$ radians about the $x$-axis. Lengths are in centimetres.

   (a)

   Find the volume of the solid.

   [1 mark]

   • A$\frac{16\pi}{3}$ cm³
   • B$16\pi$ cm³
   • C$\frac{64\pi}{3}$ cm³
   • D$8\pi$ cm³

   (b)

   Find the $x$-coordinate of the centre of mass of the solid.

   [1 mark]

   • A$2$ cm
   • B$3$ cm
   • C$\frac83$ cm
   • D$2.4$ cm

   (c)

   A second solid is formed by rotating the region bounded by $y=\sqrt{x}$, the $x$-axis and the line $x=9$ about the $x$-axis. Find the $x$-coordinate of its centre of mass.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform lamina occupies the region bounded by the curve $y=4-x^2$, the $x$-axis and the $y$-axis, for $0\le x\le2$. Lengths are in metres.

   (a)

   Show that the $x$-coordinate of the centre of mass of the lamina is $\frac34$ m.

   [3 marks]

   (b)

   Find the $y$-coordinate of the centre of mass of the lamina.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform solid hemisphere of radius $r$ is formed by rotating the region bounded by the curve $y=\sqrt{r^2-x^2}$, the $x$-axis and the $y$-axis through $2\pi$ radians about the $x$-axis.

   (a)

   Show that the centre of mass of the hemisphere is $\frac{3r}{8}$ from the centre of its plane face.

   [6 marks]

   (b)

   A toy is made from a uniform solid hemisphere of radius $6$ cm and a uniform solid cylinder of radius $6$ cm and height $8$ cm, made of the same material. The plane face of the hemisphere is joined to a circular end of the cylinder. Use the result in (a) to find the distance of the centre of mass of the toy from the joined face.

   [6 marks]

   Total for question 4: 12 marks

End of questions