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IB Maths AI SL 5.8 Trapezoidal rule

IB Maths AI SL 5.8 Trapezoidal rule
IB Mathematics: applications and interpretation guide, first assessment 2021

Students practise estimating definite integrals numerically using the trapezoidal rule with given data or function values.

How this is tested

  • apply trapezoidal rule with stated interval width h, using all ordinates from table or diagram
  • estimate cross-sectional area, then use flow rate to find volume per second
  • sum three trapezoid areas separately when board widths and heights are given in segments

Question 5(c)

[Maximum number: 3]

Zheng models the height of a section of a rollercoaster.
She uses the function

f(x)=0.001x3−0.12x2+4.05x+3, for 0≤x≤60f(x)=0.001 x^{3}-0.12 x^{2}+4.05 x+3, \text { for } 0 \leq x \leq 60

where x is the horizontal distance from the left-hand end of the section and f(x) is the vertical height. Distances are measured in metres.

Zheng's model is shown on the following graph of y=f(x).

Figure for Question 5(c) — IB Maths AI SL

Point O has coordinates (0,0).

Find the total area of the advertising boards.

The cost of space on the advertising boards per square metre varies according to the amount purchased.

The rate of change in the cost, in dollars ( $), for space on the advertising boards can be modelled by the function

C′(a)=−0.00292a+4.71, for a>60C^{\prime}(a)=-0.00292 a+4.71, \text { for } a>60

where a is the total square metres purchased.
A company, E-Byte, wishes to advertise its new gaming device. Zheng tells E-Byte it will cost $ 1750 for 500 m2500 \mathrm{~m}^{2} of space on the advertising boards.