Volumes of revolutionEdexcel A-Level Further Maths: Subtopic test

1. 1
   The region $R$ is bounded by the curve $y=\sqrt x$, the $x$-axis and the line $x=4$.

   (a)

   $R$ is rotated through $2\pi$ radians about the $x$-axis. Find the volume of the solid formed.

   [1 mark]

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

   (b)

   $R$ is rotated through $2\pi$ radians about the $y$-axis. Find the volume of the solid formed.

   [1 mark]

   • A$\frac{128\pi}{5}$
   • B$\frac{32\pi}{5}$
   • C$32\pi$
   • D$8\pi$

   (c)

   The solid formed in (a) is cut so that only the part with $1\leq x\leq4$ remains. Find the exact volume of this part.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\frac1x$ for $x>0$. All lengths are in centimetres.

   (a)

   The region bounded by $C$, the $x$-axis and the lines $x=1$ and $x=3$ is rotated through $2\pi$ about the $x$-axis. Find the volume generated, in cm$^3$.

   [1 mark]

   • A$\frac23$
   • B$\pi\ln3$
   • C$\frac{2\pi}{3}$
   • D$\frac{8\pi}{9}$

   (b)

   The region bounded by $C$, the $y$-axis and the lines $y=1$ and $y=5$ is rotated through $2\pi$ about the $y$-axis. Find the volume generated, in cm$^3$.

   [1 mark]

   • A$\pi\ln5$
   • B$\frac{24\pi}{25}$
   • C$\frac45$
   • D$\frac{4\pi}{5}$

   (c)

   A vase is modelled by rotating the part of $C$ between $x=2$ and $x=5$ through $2\pi$ radians about the $x$-axis. Find the exact volume of the vase.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The finite region $R$ is bounded by the curve $y=x^2$ and the line $y=2x$.

   (a)

   Find the exact volume generated when $R$ is rotated through $2\pi$ radians about the $x$-axis.

   [3 marks]

   (b)

   Find the exact volume generated when $R$ is rotated through $2\pi$ radians about the $y$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\sin x$ for $0\leq x\leq\pi$.

   (a)

   The region $R$ is bounded by $C$ and the $x$-axis. Show that the volume generated when $R$ is rotated through $2\pi$ radians about the $x$-axis is $\frac{\pi^2}{2}$.

   [6 marks]

   (b)

   The region $S$ is bounded by $C$ and the line $y=\frac12$. Find the exact volume generated when $S$ is rotated through $2\pi$ radians about the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions