I: Numerical methods: Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.

The idea

This topic covers numerical integration using the trapezium rule to estimate the area under a curve y=f(x) between x=a and x=b. The rule divides the interval into n strips of equal width h=(b-a)/n and sums trapezium areas: integral ≈ (h/2)[y_0 + 2(y_1+...+y_{n-1}) + y_n]. You must understand that the estimate is approximate and can be an over- or under-estimate depending on whether the curve is concave or convex over the interval. You should also be able to determine upper and lower bounds for the estimate, for example by using rectangles or considering the nature of the curve.

For example: Estimate ∫_0^1 √(1+x^3) dx using 4 strips and state whether it is an over- or under-estimate.

What earns the marks

• Applies the trapezium rule formula correctly, identifying h, the number of strips, and the ordinates y_0, y_1, ..., y_n.
• Calculates the ordinates accurately, often using a table, and substitutes them into the formula without arithmetic errors.
• Determines whether the trapezium rule gives an over- or under-estimate by considering the concavity of the curve over the interval.
• Estimates limits between which the true area must lie, for example by using inscribed and circumscribed rectangles or by comparing with other approximations.
• Interprets the result in context, including units if applicable, and recognises that increasing the number of strips generally improves the approximation.

Common mistakes and how to put them right

• Forgetting to multiply the interior ordinates by 2 in the trapezium rule.

  Put it right: The pattern y_0 + 2(y_1+...+y_{n-1}) + y_n.

• Using the wrong strip width, for example taking h=b-a instead of h=(b-a)/n.

  Put it right: Divide the interval width by the number of strips.

• Assuming the trapezium rule always underestimates.

  Put it right: Check the concavity; for a convex curve (concave up) the trapezia lie above the curve, giving an over-estimate, and vice versa.

• Rounding ordinates too early, leading to loss of accuracy.

  Put it right: Keep full precision until the final answer.

Exam tips

• Draw a sketch of the curve and the trapezia to help decide whether the estimate is an over- or under-estimate.
• Show the table of ordinates and the substitution into the formula clearly, even if you use a calculator.
• If asked for limits, consider using the minimum and maximum values of f(x) on each subinterval to bound the area.
• State the number of strips and the value of h explicitly before calculating.