The diagram shows the graph of the function $y = ext{sin} \, x$ in the domain $0 \leq x \leq \pi$, $x \in \mathbb{R}$ - Leaving Cert Mathematics - Question 6 - 2015

The trapezoidal rule formula is given by:

$A = \frac{h}{2} [y_1 + 2(y_2 + y_3 + \ldots + y_{n-1}) + y_n]$

Here, we have:

• $h = \frac{\pi}{6}$
• $y_0 = 0$, $y_1 = 0.5$, $y_2 = 0.866$, $y_3 = 1$, $y_4 = 0.866$, $y_5 = 0.5$, $y_6 = 0$

Thus,

$A = \frac{\frac{\pi}{6}}{2} [0 + 2(0 + 0.5 + 0.866 + 1 + 0.866 + 0.5)] = \frac{\pi}{12} [0 + 2 \times 3.732] \approx 1.95407$

To find the actual area, we integrate $y = \text{sin} \, x$ from $0$ to $\pi$:

$\int_0^{\pi} \text{sin} \, x \, dx = [-\text{cos} \, x]_0^{\pi} = -[-1 - 1] = 2$

The percentage error can be calculated as follows:

$\text{Percentage error} = \left(\frac{|\text{Actual Area} - \text{Trapezoidal Area}|}{\text{Actual Area}}\right) \times 100 = \left(\frac{|2 - 1.95407|}{2}\right) \times 100 \approx 2.3\%$