Volumes of revolution and mean valueAQA A-Level Further Maths: Subtopic test

1. 1
   The region $R$ is bounded by the curve $y=x^2$, the $x$-axis and the line $x=2$. Lengths are in centimetres.

   (a)

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

   [1 mark]

   • A$\dfrac{8\pi}{3}$ cm$^3$
   • B$32\pi$ cm$^3$
   • C$8\pi$ cm$^3$
   • D$\dfrac{32\pi}{5}$ cm$^3$

   (b)

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

   [1 mark]

   • A$16\pi$ cm$^3$
   • B$8\pi$ cm$^3$
   • C$\dfrac{32\pi}{5}$ cm$^3$
   • D$4\pi$ cm$^3$

   (c)

   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]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3x^2-2x$ for $0\le x\le2$.

   (a)

   Find the mean value of $\mathrm{f}$ over the interval $0\le x\le2$.

   [1 mark]

   • A$2$
   • B$4$
   • C$8$
   • D$1$

   (b)

   Find the value of $x$ in the interval $0\le x\le2$ at which $\mathrm{f}(x)$ equals the mean value of $\mathrm{f}$ over that interval.

   [1 mark]

   • A$\dfrac{1-\sqrt{7}}{3}$
   • B$1$
   • C$\dfrac{1+\sqrt{7}}{3}$
   • D$\dfrac{1+\sqrt{13}}{3}$

   (c)

   Find the mean value of $\mathrm{f}$ over the interval $1\le x\le2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The region $R$ is bounded by the curve $y=\dfrac{1}{\sqrt{2x+1}}$, the coordinate axes and the line $x=4$.

   (a)

   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]

   (b)

   The line $x=4$ is replaced by the line $x=a$, where $a>0$. The new region is rotated through $2\pi$ radians about the $x$-axis. Given that the volume of the solid formed is $2\pi\ln3$, find the value of $a$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_1$ with equation $y=x^2$ and the curve $C_2$ with equation $y=\sqrt{x}$ intersect at the origin and at one other point. The region $R$ is enclosed between $C_1$ and $C_2$. Lengths are in metres.

   (a)

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

   [6 marks]

   (b)

   $R$ is now rotated through $2\pi$ radians about the $y$-axis. Find the volume of the solid formed, and explain, without any further integration, why it is equal to your answer to (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions