Split where the mathematics changes
For modulus integrals, locate every zero of the inner expression and change sign piece by piece. For floor or ceiling extensions, locate the jump points and integrate a constant on each subinterval. For an unknown function, transform the limits carefully before trying to infer anything about the integrand itself.
A definite integral is not automatically an area. If the graph crosses the axis, signed contributions can cancel; if the question asks for geometric area, the sign must be handled explicitly.
Trapezium rule questions are geometric
A convex-up curve lies below its secant segments, so the trapezium rule overestimates; a concave-down curve gives the opposite behaviour. For a piecewise-linear modulus graph, a grid containing every breakpoint can make the trapezium rule exact.
Do not assume a finer grid is exact simply because it is finer. What matters is how the grid aligns with the structure of the function.
FREE PRACTICE
Try these TMUA-style questions
These questions now draw from the current reviewed CF Maths banks, including image and diagram items where appropriate. Open the solution only after you have committed to a route.
QUESTION 1 · Trapezium rule & numerical integrationShow answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Trapezium rule & numerical integration. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Definite integrals & parameters. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Areas & absolute-value integrals. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Floor/ceiling integrals. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: D
Correct answer: \(D\). This item is marked from the validated final 200-question answer key. Revision method: Trapezium rule & numerical integration. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Definite integrals & parameters. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Definite integrals & parameters. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Definite integrals & parameters. Rework the displayed question carefully and compare your algebra with the keyed option.
