Volumes of revolutionEdexcel International A Level Maths: Mind map

Exam questions on Volumes of revolution

1. The region $R$ is bounded by the curve $y=2\sqrt{x}$, the $x$-axis and the lines $x=1$ and $x=4$. The region $R$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.
   The line $x=4$ is replaced by the line $x=a$, where $a>1$. The volume of the new solid is $48\pi$. Find the value of $a$.2 marks
2. The region $S$ is bounded by the curve $y=e^{2x}$, the coordinate axes and the line $x=1$. The region $S$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.
   The line $x=1$ is replaced by the line $x=\ln2$. Find the exact volume of the new solid.2 marks
3. The region $R$ is bounded by the curve $y=\frac{1}{\sqrt{2x+1}}$, the coordinate axes and the line $x=4$. The region $R$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.
   Show that the volume of the solid is $\pi\ln3$.3 marks