Oxford University MAT 2015 Solved Papers

F. Sketching $\mathit{f}\left(\mathit{x}\right)$ produces a saw-tooth curve (a series of right-angled triangles).
The integral is equal to $\frac{2\times 1}{2}+\frac{2\times 1}{2}+\frac{0.5\times 1}{2}=2.25$ by inspection. 2 strips produces an underestimate (either by inspection or by calculation of the area of the resulting triangle $\frac{5\times 0.5}{2}=1.25$ ). 3 strips produces an overestimate - either by inspection or by algebra. Either using the trapezium rule, or by breaking the area down into the first triangle (area $=\frac{5}{3}\times \frac{5}{6}\times \frac{1}{2}=\frac{25}{36})$ and the following part triangle $($ area $=\left(\frac{25}{3}\times \frac{5}{6}\times \frac{1}{2}\right)-\left(\frac{15}{3}\times \frac{1}{2}\times \frac{1}{2}\right)=\frac{80}{36})$. Since $\frac{105}{36}>\frac{81}{36}$ the answer is (b).