Volumes of revolution and mean valueAQA A-Level Further Maths: Mind map

Exam questions on Volumes of revolution and mean value

1. The region $R$ is bounded by the curve $y=x^2$, the $x$-axis and the line $x=2$. Lengths are in centimetres.
   Only the part of $R$ with $1\le x\le2$ is rotated through $2\pi$ radians about the $x$-axis. Find the exact volume of the solid formed.2 marks
2. The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3x^2-2x$ for $0\le x\le2$.
   Find the mean value of $\mathrm{f}$ over the interval $1\le x\le2$.2 marks
3. The region $R$ is bounded by the curve $y=\dfrac{1}{\sqrt{2x+1}}$, the coordinate axes and the line $x=4$.
   Show that when $R$ is rotated through $2\pi$ radians about the $x$-axis, the volume of the solid formed is $\pi\ln3$.3 marks