| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Mean value using inverse trig integral |
| Difficulty | Challenging +1.8 This Further Maths question requires completing the square to recognize inverse trig integrals (arcsin), computing a volume of revolution with a rational integrand, and evaluating an improper integral with exponential decay. While the techniques are standard for Further Maths (arcsin integration, substitution for the improper integral), the multi-step nature, algebraic manipulation required, and combination of advanced calculus topics place it well above average difficulty. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands4.08d Volumes of revolution: about x and y axes4.08e Mean value of function: using integral |
\begin{enumerate}[label=(\alph*)]
\item A curve C is defined by the equation $y = \frac{1}{\sqrt{16-6x-x^2}}$ for $-3 \leq x \leq 1$.
\begin{enumerate}[label=(\roman*)]
\item Find the mean value of $y = \frac{1}{\sqrt{16-6x-x^2}}$ between $x = -3$ and $x = 1$. [4]
\item The region $R$ is bounded by the curve C, the $x$-axis and the lines $x = -3$ and $x = 1$. Find the volume of the solid generated when $R$ is rotated through four right-angles about the $x$-axis. [5]
\end{enumerate}
\item Evaluate the improper integral
$$\int_1^{\infty} \frac{8e^{-2x}}{4e^{-2x} - 5} \mathrm{d}x,$$
giving your answer correct to three decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2024 Q7 [12]}}