The curves C1 and C2 have equations C1: y=4x−x² and C2: y = x. They intersect at x = 0 and x = 3. The region R is enclosed by C1 and C2. (a)(i) Use integration to find the exact area of R.

The curves are $C_1: y_1 = 4x - x^2$ and $C_2: y_2 = x$.
They intersect at $x=0$ and $x=3$.
To determine which curve is above the other in the interval $[0, 3]$, we can test a point, for example, $x=1$:
For $C_1: y_1(1) = 4(1) - (1)^2 = 3$.
For $C_2: y_2(1) = 1$.
Since $y_1(1) > y_2(1)$, $C_1$ is the upper curve and $C_2$ is the lower curve in the interval $[0, 3]$.

The area $A$ between the curves is given by the integral of the difference between the upper and lower curves:

Part (a)(ii) Exact Volume of the Solid Formed

The volume $V$ of the solid formed by rotating the region between two curves $y_1(x)$ and $y_2(x)$ about the x-axis is given by the washer method:

Here, $y_1 = 4x - x^2$ (outer radius) and $y_2 = x$ (inner radius), with limits $a=0$ and $b=3$.

Part (b)(i) Estimate the Area of R using the Trapezium Rule

The interval is $[0, 3]$. With 4 ordinates, there are $n=3$ strips.
The width of each strip is $h = \frac{b-a}{n} = \frac{3-0}{3} = 1$.
The x-values for the ordinates are $x_0=0, x_1=1, x_2=2, x_3=3$.
The function to integrate is $f(x) = 3x - x^2$.

The trapezium rule formula is $A_{trap} = \frac{h}{2} [y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})]$.

Part (b)(ii) Calculate the Percentage Error

Exact Area $A = \frac{9}{2} = 4.5$.
Estimated Area $A_{trap} = 4$.

Percentage error $= \left| \frac{\text{Estimated Area} - \text{Exact Area}}{\text{Exact Area}} \right| \times 100\%$
Percentage error $= \left| \frac{4 - 4.5}{4.5} \right| \times 100\%$
Percentage error $= \left| \frac{-0.5}{4.5} \right| \times 100\%$
Percentage error $= \frac{0.5}{4.5} \times 100\%$
Percentage error $= \frac{1}{9} \times 100\%$
Percentage error $= \boxed{\frac{100}{9}\% \approx 11.1\% \text{ (to 3 s.f.)}}$