2

2. [Maximum mark: 15]

The following diagram shows a model of the side view of a water slide. All lengths are measured in metres.

The curved edge of the slide is modelled by
$f(x) = -\frac{1}{4}x^{2} + 2x$ for $0 \le x \le 4$.

The remainder of the slide is modelled by
$g(x) = \begin{cases} 4, & 4 \le x \le 5 \\ \frac{48}{7} - \frac{4x}{7}, & 5 \le x \le 12 \end{cases}$

(a) Use the trapezoidal rule with an interval width of 1 to calculate the approximate area under the model of the slide in the interval $0 \le x \le 4$. [5]

(b) Find $\int \bigl(-\frac{1}{4}x^{2} + 2x\bigr)\,dx$. [3]

(c) Calculate the exact area under the entire model of the slide, for $0 \le x \le 12$. [4]

(d) Find the percentage error in the total area under the entire model of the slide when using the approximate value from part (a). [3]