Trapezium rule symmetry argument

A question is this type if and only if it involves a function that is symmetric or nearly symmetric over the interval of integration, and asks the student to explain why the trapezium rule gives a good approximation based on this symmetry.

2 questions · Moderate -0.6

1.09f Trapezium rule: numerical integration
Sort by: Default | Easiest first | Hardest first
CAIE P3 2005 June Q2
4 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-2_508_586_450_776} The diagram shows a sketch of the curve \(y = \frac { 1 } { 1 + x ^ { 3 } }\) for values of \(x\) from - 0.6 to 0.6 .
  1. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { - 0.6 } ^ { 0.6 } \frac { 1 } { 1 + x ^ { 3 } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.
CAIE P3 2017 November Q1
4 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{746d2c39-7d78-4478-bc36-15ea5e3ba72a-02_460_807_258_667} The diagram shows a sketch of the curve \(y = \frac { 3 } { \sqrt { } \left( 9 - x ^ { 3 } \right) }\) for values of \(x\) from - 1.2 to 1.2 .
  1. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { - 1.2 } ^ { 1.2 } \frac { 3 } { \sqrt { \left( 9 - x ^ { 3 } \right) } } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.