Standard +0.3 This is a standard area-between-curves integration problem requiring students to set up and evaluate ∫₀^π [(-x² + πx + 1) - cos 2x] dx. While it involves multiple terms and requires careful integration (including a trigonometric function), it's a routine application of A-level integration techniques with no conceptual surprises. The 9 marks reflect the algebraic work rather than difficulty, making it slightly easier than average.
The diagram below shows a sketch of the curve \(C_1\) with equation \(y = -x^2 + \pi x + 1\) and a sketch of the curve \(C_2\) with equation \(y = \cos 2x\). The curves intersect at the points where \(x = 0\) and \(x = \pi\).
\includegraphics{figure_9}
Calculate the area of the shaded region enclosed by \(C_1\), \(C_2\) and the \(x\)-axis. Give your answer in terms of \(\pi\). [9]
The diagram below shows a sketch of the curve $C_1$ with equation $y = -x^2 + \pi x + 1$ and a sketch of the curve $C_2$ with equation $y = \cos 2x$. The curves intersect at the points where $x = 0$ and $x = \pi$.
\includegraphics{figure_9}
Calculate the area of the shaded region enclosed by $C_1$, $C_2$ and the $x$-axis. Give your answer in terms of $\pi$. [9]
\hfill \mbox{\textit{WJEC Unit 3 2024 Q9 [9]}}