Q333 · Practice questionTechnology-freeComplex unfamiliar7 marks
QUESTION 333 (7 marks)
A curved garden border has height metres for .
![Profile f(x)=5+cos(pi x/4) on [0,4], with vertical strip boundaries at integer x and dashed straight trapezoidal upper edges.](/qce-data/assets/methods-2026-MM26-ORIG-095-mobile.svg)
a)[2 marks]
Determine its exact area above the baseline.
b)[2 marks]
Use four equal strips to calculate a trapezoidal estimate exactly.
c)[3 marks]
Explain why the trapezoidal rule gives the exact area for any positive integer number of equal strips in this particular example.
WORKED SOLUTION
7 marksPractice marking scheme
ANSWER
a) . b) . c) Symmetric cosine contributions cancel; the constant part is integrated exactly.
Worked solution
a) .
b) The heights are , , , , . The trapezoidal sum is .
c) At paired nodes and , . Their trapezoidal weights are equal, including half-weight endpoints, so every cosine contribution cancels (a possible central node contributes zero). The constant contributes .
Exact answers unless rounding is specified; equivalent forms accepted.
a) Integrates the profile.
a) Evaluates the exact area.
b) Gives the exact node heights.
b) Applies the trapezoidal weights.
c) Establishes cancellation at symmetric nodes.
c) Accounts for equal weights and the central node.
c) Explains the exact contribution of the constant term.
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