Given that $y = ext{sec} \, x$, complete the table with the values of $y$ corresponding to $x = \frac{\pi}{16}, \frac{\pi}{8}$ and $\frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 4 - 2006 - Paper 7

To estimate the integral using the trapezium rule, we use the formula:

$I \approx \frac{b - a}{2n} [f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b)]$

Where:

• $n = 4$ (the number of segments)
• $a = 0$, $b = \frac{\pi}{4}$
• $f(x) = \text{sec} \, x$

Thus, the integral can be estimated as:

$I \approx \frac{\frac{\pi}{4} - 0}{2 imes 4} [1 + 2(1.01959 + 1.08239 + 1.20269) + 1.41421]$

Calculating the sum: $I \approx \frac{\frac{\pi}{4}}{8} [1 + 2(1.01959 + 1.08239 + 1.20269) + 1.41421] \approx \frac{\frac{\pi}{4}}{8} [1 + 2(3.30467) + 1.41421]$

Then simplifying: $I \approx \frac{\frac{\pi}{4}}{8} [1 + 6.60934 + 1.41421] \approx \frac{\frac{\pi}{4}}{8} [8.02355] \approx 0.8859$

Thus, the estimate for the integral is approximately $0.8859$.

The exact value of the integral is given as:

$\int_0^{\frac{\pi}{4}} \text{sec} \, x \, dx = \ln(1 + \sqrt{2})$

Using a calculator, we find that: $\approx 0.88137$

Now calculating the percent error using the formula:

$\text{Percentage Error} = \frac{\text{approx} - \text{exact}}{\text{exact}} \times 100$

Substituting the values: $\text{Percentage Error} = \frac{0.8859 - 0.88137}{0.88137} \times 100 \approx 0.51\%$

Thus, the % error in the estimate obtained in part (b) is approximately $0.51\%$.