CAIE A-Level Further Math 2.4.3 How the area under a curve

$\int_{0}^{1} \cos x \mathrm{~d} x<\frac{1}{n}+\frac{1}{n} \cos \frac{1}{n}+\frac{1}{n} \cos \frac{2}{n}+\ldots \frac{1}{n} \cos \frac{n-1}{n}$

M1 A1

Forms sum of areas of rectangles given in the diagram.
A0 if comparison with integral missing or unclear.

$=\frac{1}{n}\left(1+\cos \frac{1}{n}+\cos \frac{2}{n}+\ldots+\cos \frac{n-1}{n}\right)=\frac{1}{2 n}\left(1-\cos \frac{n}{n}+\frac{\sin \frac{n}{n} \sin \frac{1}{n}}{1-\cos \frac{1}{n}}\right)$

M1 A1

Applies result from part (b) with $\theta=\frac{1}{n}$. AG.

4

$\int_{0}^{1} \cos x \mathrm{~d} x>\frac{1}{n} \cos \frac{1}{n}+\frac{1}{n} \cos \frac{2}{n}+\ldots \frac{1}{n} \cos \frac{n}{n}$

M1 A1

Forms sum of areas of rectangles. A0 if comparison
with integral missing or unclear.

$=\frac{1}{2 n}\left(1-\cos 1+\frac{\sin 1 \sin \frac{1}{n}}{1-\cos \frac{1}{n}}\right)+\frac{1}{n} \cos 1-\frac{1}{n}=\frac{1}{2 n}\left(\cos 1-1+\frac{\sin 1 \sin \frac{1}{n}}{1-\cos \frac{1}{n}}\right)$

A1

3