1. 1
   The function $f$ is defined by $f(x) = 3x^{2} + 2x$ for $x \in \mathbb{R}$.

   (a)

   Find the value of $\displaystyle\int_{1}^{2} f(x)\,\mathrm{d}x$.

   [1 mark]

   • A$11$
   • B$27$
   • C$10$
   • D$12$

   (b)

   Find the value of $\displaystyle\int_{1}^{2} \left(f(x) + 4\right)\mathrm{d}x$.

   [1 mark]

   • A$18$
   • B$14$
   • C$10$
   • D$40$

   (c)

   Find the value of $\displaystyle\int_{-1}^{0} f(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x) = \dfrac{4}{2x+1}$ for $x \ge 0$. The function $h$ satisfies $h'(x) = g(x)$ for $x \ge 0$, and $h(0) = 3$.

   (a)

   Find the exact value of $\displaystyle\int_{0}^{4} g(x)\,\mathrm{d}x$.

   [1 mark]

   • A$2\ln 9$
   • B$4\ln 9$
   • C$-\dfrac{32}{9}$
   • D$8\ln 9$

   (b)

   Find the exact value of $h(4)$.

   [1 mark]

   • A$2\ln 9$
   • B$3 - 2\ln 9$
   • C$3 + 4\ln 9$
   • D$3 + 2\ln 9$

   (c)

   Given that $\displaystyle\int_{0}^{k} g(x)\,\mathrm{d}x = 2\ln 5$, where $k > 0$, find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The rate of flow of water into a reservoir is modelled by $r(t) = 6t - t^{2}$, where $r$ is measured in thousands of cubic metres per day and $t$ is the time in days, for $0 \le t \le 8$. When $r(t) < 0$, water is flowing out of the reservoir. Do not use a calculator in this question.

   (a)

   Find the net change in the volume of water in the reservoir over the 8 days.

   [3 marks]

   (b)

   Find the total volume of water that flowed out of the reservoir during the 8 days, and the total volume that flowed in. Hence verify your answer to part (a).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y = x^{3} - 4x$ and the line $L$ has equation $y = 5x$. Do not use a calculator in this question.

   (a)

   (i) Show that $\displaystyle\int_{-2}^{2} (x^{3} - 4x)\,\mathrm{d}x = 0$.
   (ii) Explain why this integral does not give the area of the region enclosed by $C$ and the $x$-axis.
   (iii) Find the total area of the region enclosed by $C$ and the $x$-axis.

   [6 marks]

   (b)

   Find the total area of the two regions enclosed between $C$ and $L$.

   [6 marks]

   Total for question 4: 12 marks

End of questions