Centres of mass of uniform bodies by integrationEdexcel International A Level Further Maths: Mind map

Exam questions on Centres of mass of uniform bodies by integration

1. A uniform lamina occupies the region $R$ bounded by the curve $y=x^2$, the $x$-axis and the line $x=2$.
   Find the $y$-coordinate of the centre of mass of $R$.2 marks
2. A uniform solid $S$ is formed by attaching a solid hemisphere of radius $4$ cm to one end of a solid cylinder of radius $4$ cm and height $10$ cm. The hemisphere and the cylinder are made of the same material, and the plane face of the hemisphere coincides with an end face of the cylinder.
   Find the distance of the centre of mass of $S$ from the end face of the cylinder that is not in contact with the hemisphere.2 marks
3. A uniform solid $S$ is formed by rotating the region bounded by the curve $y=\sqrt x$, the $x$-axis and the line $x=4$ through $2\pi$ radians about the $x$-axis. The units are centimetres.
   Use integration to show that the centre of mass of $S$ is $\frac83$ cm from the origin $O$.3 marks