Use the trapezoidal rule, ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 ≈ 𝑤 2 [𝑓(𝑥0) + 2(𝑓(𝑥1) + 𝑓(𝑥2) + 𝑓(𝑥3)+... 𝑓(𝑥𝑛−1)) + 𝑓(𝑥𝑛)], where 𝑤 = 𝑏−𝑎 𝑛, to approximate an area and the value of a definite integral, with and without technology.

State the trapezoidal rule and use it with five strips to determine an approximate value of the definite integral for the curve of h(x) = 3(x - 2)² + 1 from x = 0 to x = 5. Show all substitutions made into the rule.

A function is defined by \( f(x) = \sqrt{x^2 + 4} \). (a) State the trapezoidal rule for approximating a definite integral. [1 mark] (b) Use the trapezoidal rule with five strips to determine an approximate value for \( \int_{1}^{6} \sqrt{x^2 + 4} \, dx \). Show all substitutions made into the rule. [3 marks]

The velocity, \( v \) (m/s), of a cyclist during a 12-second interval is recorded at regular intervals as shown in the graph below. Using the trapezoidal rule with four subintervals, the distance travelled by the cyclist during the 12-second interval is approximately