| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.8 This is a straightforward C2 numerical methods question requiring basic function evaluation, standard trapezium rule application with given ordinates, and simple area subtraction. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to multi-step calculation requirements. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 0 | 0.530 | 6 |
The curve $C$ has equation
$y = x\sqrt{x^2 + 1}, \quad 0 \leq x \leq 2$.
\begin{enumerate}[label=(\alph*)]
\item Copy and complete the table below, giving the values of $y$ to 3 decimal places at $x = 1$ and $x = 1.5$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
$y$ & 0 & 0.530 & & & 6 \\
\hline
\end{tabular}
\end{center}
[2]
\item Use the trapezium rule, with all the $y$ values from your table, to find an approximation for the
value of $\int_0^2 x\sqrt{x^2 + 1} \, dx$, giving your answer to 3 significant figures.
[4]
\end{enumerate}
\includegraphics{figure_2}
Figure 2 shows the curve $C$ with equation $y = x\sqrt{x^2 + 1}$, $0 \leq x \leq 2$, and the straight line segment $l$, which joins the origin and the point $(2, 6)$. The finite region $R$ is bounded by $C$ and $l$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use your answer to part (b) to find an approximation for the area of $R$, giving your answer to 3 significant figures.
[3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [9]}}