Understand and use numerical integration of functions, including the use…

The idea

Numerical integration estimates a definite integral when an antiderivative is unavailable or inconvenient. The trapezium rule replaces the area under y = f(x) between x = a and x = b with n strips of equal width h = (b − a) ÷ n. Each strip is a trapezium, so the total is h/2 × [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]. For example, with h = 0.5, y₀ = 1, y₁ = 2, y₂ = 5, the estimate is 0.5/2 × [1 + 5 + 2(2)] = 2.5. A curve that is concave up lies below its chords, so the rule overestimates; concave down gives an underestimate. The true area therefore lies between the trapezium estimate and a bound obtained by considering the curve's curvature, and increasing n generally tightens that interval.

What earns the marks

• States the trapezium rule correctly as h/2 × [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)] with h = (b − a) ÷ n and uses it to estimate a definite integral.
• Computes strip width h accurately from the limits and number of strips, then tabulates ordinates y₀ to yₙ at x = a, a + h, …, b, showing sufficient working.
• Explains that the estimate is approximate because straight chords replace the curve, and links the direction of error to whether the curve is concave up or concave down over the interval.
• Determines limits between which the true area must lie, for example by comparing the trapezium estimate with an inscribed or circumscribed bound, and interprets the result in context.
• Uses increased n or decreased h to improve the estimate and comments on convergence towards the true area, recognising that the rule is exact for linear functions.

Common mistakes and how to put them right

• Doubling the first and last ordinates instead of only the interior ordinates.

  Put it right by applying the factor 2 solely to y₁ through yₙ₋₁.

• Using h = b − a rather than h = (b − a) ÷ n.

  Put it right by dividing the interval width by the number of strips before substituting.

• Assuming the trapezium rule always underestimates.

  Put it right by checking concavity, since concave-up curves give overestimates and concave-down curves give underestimates.

Exam tips

• Write down h, the ordinate values and the bracket sum separately before evaluating, so arithmetic slips are easy to spot and method is visible.
• When asked for limits, give a lower and an upper value with a brief justification based on curvature rather than quoting the estimate alone.
• Check whether the question specifies the number of ordinates or strips; if ordinates are given, n is one less than the number of ordinates.