| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Rectangle bounds for definite integral |
| Difficulty | Moderate -0.3 This is a straightforward application of upper and lower Riemann sums using rectangles to bound an area. While it's FP2, the question requires only basic understanding of how rectangles approximate areas (using left/right endpoints for bounds) and simple calculator work to evaluate ln(cos x) at given points. The conceptual demand is minimal—standard textbook exercise with clear instructions and diagram. |
| Spec | 1.09f Trapezium rule: numerical integration |
1\\
\includegraphics[max width=\textwidth, alt={}, center]{cf77e51a-1d3f-423a-be59-96ec60fbeb67-2_568_959_269_593}
The diagram shows the curve with equation $y = \ln ( \cos x )$, for $0 \leqslant x \leqslant 1.5$. The region bounded by the curve, the $x$-axis and the line $x = 1.5$ has area $A$. The region is divided into five strips, each of width 0.3 .\\
(i) By considering the set of rectangles indicated in the diagram, find an upper bound for $A$. Give the answer correct to 3 decimal places.\\
(ii) By considering another set of five suitable rectangles, find a lower bound for $A$. Give the answer correct to 3 decimal places.\\
(iii) How could you reduce the difference between the upper and lower bounds for $A$ ?
\hfill \mbox{\textit{OCR FP2 2009 Q1 [5]}}