June 2023 Paper 3 Q12

12

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Approximating series” are reproduced below; the line numbers are those printed on the Insert.

Lines 17–19
Euler’s formula relates a sum of terms to an integral, and this can be illustrated by considering a suitable graph. For the sum of the squares of natural numbers, this is the graph of \(y = x^2\). The diagram shows this curve, with four shaded rectangles of areas \(1^2\), \(2^2\), \(3^2\) and \(4^2\).

Lines 20–21
Euler’s approximate formula for this case is as follows.
\(\displaystyle\sum_{r=1}^{4} r^2 \approx \int_1^4 x^2\,\mathrm{d}x + \frac{4^2 + 1^2}{2} + \frac{1^2 - 2^2}{12} - \frac{4^2 - 5^2}{12}\)

Lines 22–25
The integral gives the area under the curve between \(x = 1\) and \(x = 4\). It is clear that the integral is smaller than \(\sum_{r=1}^{4} r^2\) so something needs to be added to the integral to get the same answer as the series. The rectangle for \(1^2\) needs to be added on and so do the parts of the other three rectangles that are above the curve.

Lines 26–27
Approximating the curve by a series of straight lines gives three triangles to be added on. These have areas \(\dfrac{2^2 - 1^2}{2}\), \(\dfrac{3^2 - 2^2}{2}\) and \(\dfrac{4^2 - 3^2}{2}\).

Lines 28–30
This gives an approximation for the series of \(\displaystyle\int_1^4 x^2\,\mathrm{d}x + 1^2 + \frac{2^2 - 1^2}{2} + \frac{3^2 - 2^2}{2} + \frac{4^2 - 3^2}{2}\) which simplifies to \(\displaystyle\int_1^4 x^2\,\mathrm{d}x + \frac{4^2 + 1^2}{2}\). The final two terms in Euler’s approximate formula are to correct for the curve not being a series of straight lines.

With the aid of a suitable diagram, show that the three triangles referred to in line 26 have the areas given in line 27. [3]