NCEA 91579: apply integration methods (Level 3 Calculus)

What NCEA 91579 assesses

NZQA's standard lists the methods:

• integrating power, polynomial, exponential (base e only), trigonometric and rational functions
• the reverse chain rule and trigonometric formulae
• rates of change problems
• areas under or between graphs by integration, and areas by numerical methods such as the rectangle or trapezium rule
• differential equations of the forms y′ = f(x) or y″ = f(x), or with separable variables such as y′ = ky, in growth and decay, inflation and Newton's Law of Cooling

Achieved selects and uses methods. Merit uses relational thinking: a logical sequence of steps, connecting representations, and relating findings to a context. Excellence uses extended abstract thinking: devising a strategy, a chain of logical reasoning or proof, or a generalisation. NZQA's specification adds that solving differential equations may require manipulating exponents and logarithms.

The NCEA 91579 exam in 2026 and 2027

NCEA 91579 is one of three Level 3 Calculus externals, with 91577 (complex numbers) and 91578 (differentiation). The 2026 exam is on Monday 16 November, in the morning session. NZQA's 2027 timetable is not out yet: it says exams run from 9 November to 3 December 2027. The 2026 and 2027 specifications match: printed paper, a Formulae and Tables Booklet, and an approved calculator, preferably a graphing calculator.

• Show the integral. NZQA says a graphing calculator answer gets no credit unless you also give the correct integrated function: you must show the result of any integration needed.
• Form your own equations at Merit and Excellence, for example a differential equation from a description.
• Round late. Three significant figures is enough unless told otherwise.

How NCEA 91579 is marked: grade score marking

Each of three questions scores N0 to E8, giving a total out of 24. Showing the integrated function protects your u units, and a correctly modelled differential equation or area problem supplies the r and t units.

NCEA 91579 cut scores

These are NZQA's cut scores for the total out of 24, taken from each year's assessment schedule. Over these five years Achieved needed 7 or 8, Merit 13 to 15, and Excellence 19 to 21. In the 2023 schedule NZQA prints the Merit range as "13 to 8", a typo: with Excellence starting at 19, Merit ran from 13 to 18. The 2026 cut scores will appear in NZQA's 2026 schedule once the exam has been marked.

Worked examples at each grade

Achieved: the reverse chain rule

Find the value of the integral below. The factor 6x is the derivative of x² + 1 multiplied by 3, so reverse the chain rule:

Write the integrated function down. That is the evidence NZQA asks for, and a calculator's final number alone is not.

Merit: a differential equation in context

A drink cools according to dT/dt = −k(T − 20), where T is its temperature in °C and t is minutes. It starts at 90 °C and is 70 °C after 10 minutes. Separate and integrate to get T − 20 = A e^(−kt), so with A = 70:

Merit asks you to link the steps and interpret the answer: after half an hour the drink is about 45.5 °C, and it will approach 20 °C, the room temperature, as t grows.

Excellence: area with an unknown constant

The region between y = x² and y = kx (k > 0) has area 36 square units. Find k. The curves meet at x = 0 and x = k, and the line is above the curve between them:

Excellence evidence is devising the strategy yourself: finding the limits in terms of k, setting up the area, and solving for the constant. The specification says questions may use unknown constants.

Check your integration with a question on the page:

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the daily paper as a practice exam

Uni-Sine publishes a free daily paper on the NCEA Level 3 Calculus track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. Its topics include integration techniques, area under a curve, differential equations, and integration by substitution and by parts, which cover most of 91579. Write the integrated function in every answer, as the exam requires. See also NCEA 91578 and the NCEA Level 3 Calculus guide.

Common mistakes in NCEA 91579

• Calculator-only answers. No integrated function, no credit.
• Forgetting the constant. Add + c for an indefinite integral, and use the given condition to find it.
• Area signs. Area below the x-axis integrates to a negative number, so split the integral at the roots.
• Skipping context. State what a rate, area or temperature answer means with its units.

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