Centre of mass by integrationAQA A-Level Further Maths: Mind map

Exam questions on Centre of mass by integration

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 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
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 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
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.
   Show that the $x$-coordinate of the centre of mass of the lamina is $\frac34$ m.3 marks