1. 1
   The curve $C$ has equation $y=\ln x$, for $x>0$. The region $R$ is enclosed by $C$, the $y$-axis, the $x$-axis and the line $y=2$.

   (a)

   Which of the following gives the area of $R$?

   [1 mark]

   • A$\displaystyle\int_{0}^{2} e^{y}\,dy$
   • B$\displaystyle\int_{1}^{e^{2}} \ln x\,dx$
   • C$\displaystyle\int_{0}^{2} \ln y\,dy$
   • D$\displaystyle\int_{0}^{e^{2}} e^{y}\,dy$

   (b)

   Find the area of $R$.

   [1 mark]

   • A$e^{2}$
   • B$e^{2}+1$
   • C$2e^{2}$
   • D$e^{2}-1$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The region $S$ is enclosed by the curve $y=\dfrac{2}{\sqrt{x}}$, the $x$-axis and the lines $x=1$ and $x=4$. The region $S$ is rotated through $2\pi$ radians about the $x$-axis to form a solid of volume $V$.

   (a)

   Which of the following is equal to $V$?

   [1 mark]

   • A$\pi\displaystyle\int_{1}^{4}\frac{2}{\sqrt{x}}\,dx$
   • B$\displaystyle\int_{1}^{4}\frac{4}{x}\,dx$
   • C$\pi\displaystyle\int_{1}^{4}\frac{4}{x}\,dx$
   • D$\pi\displaystyle\int_{1}^{4}\frac{4}{x^{2}}\,dx$

   (b)

   Find the exact value of $V$.

   [1 mark]

   • A$4\pi$
   • B$3\pi$
   • C$8\pi\ln 2$
   • D$2\pi\ln 2$

   (c)

   The part of $S$ between $x=1$ and $x=k$, where $1<k<4$, is rotated through $2\pi$ radians about the $x$-axis. The solid formed has volume $\dfrac{V}{2}$. Find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A glass bowl is modelled by rotating the part of the curve $y=\dfrac{x^{2}}{4}$ for which $0\le y\le 9$ through $2\pi$ radians about the $y$-axis. All lengths are in centimetres. The bowl stands on a horizontal table with its axis vertical, and water is poured in until the depth of water, measured from the lowest point of the bowl, is $h$ cm.

   (a)

   Show that the volume of water in the bowl is $2\pi h^{2}$ cm$^{3}$.

   [3 marks]

   (b)

   The bowl is filled with water to exactly half of its total capacity. Find the depth of the water, giving your answer to three significant figures, and explain why this depth is more than half the height of the bowl.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curves $y=\sqrt{x}$ and $y=\dfrac{x}{2}$ meet at the origin and at one other point. The region $R$ is the finite region enclosed between the two curves.

   (a)

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

   [6 marks]

   (b)

   (i) By integrating with respect to $y$, find the area of $R$.
   (ii) The region $R$ is rotated through $2\pi$ radians about the $y$-axis. Find the exact volume of the solid formed.

   [6 marks]

   Total for question 4: 12 marks

End of questions