Standard +0.3 This is a standard area-between-curves question requiring students to find intersection points, set up integrals, and integrate two functions (a square root and an exponential). The algebra is straightforward: both curves pass through (0,3), and the integrals are routine A-level techniques. The final answer manipulation to get the form p + q ln 2 is also standard. This is slightly easier than average because it's a well-practiced question type with no conceptual surprises.
In this question you must show detailed reasoning.
\includegraphics{figure_6}
The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis.
Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]
\textbf{In this question you must show detailed reasoning.}
\includegraphics{figure_6}
The diagram shows the curves $y = \sqrt{2x + 9}$ and $y = 4\mathrm{e}^{-2x} - 1$ which intersect on the $y$-axis. The shaded region is bounded by the curves and the $x$-axis.
Determine the area of the shaded region, giving your answer in the form $p + q \ln 2$ where $p$ and $q$ are constants to be determined. [8]
\hfill \mbox{\textit{OCR H240/03 2022 Q6 [8]}}