Definite Integrals and Area NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

Choose a question, attempt it independently, then compare the reasoning and restrictions in its walkthrough.

Relevant standards: NCEA Level 3 Integration — AS91579

This collection contains 45 matching walkthroughs across 9 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

Find the correct bounds and distinguish signed integrals from geometric area. Differentiate your antiderivative before evaluating it.

Related practice

Definite Integrals and Area questions from different years

Open a question for hints and a progressive worked solution, or use the standard and paper links to return to its assessment context.

NCEA Level 3 · AS91579 2025 Integration 7 questions

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Question 1(c) — Definite Integral Equations Linking two definite integrals to solve for constants. Question 1(d) — Integration and Area Using product-to-sum before integrating a shaded trig area. Question 2(c) — Definite Integral Equations Using a definite integral to determine a constant. Question 2(e) — Integration by Substitution Substitution in a shaded trigonometric area problem. Question 3(a) — Numerical Integration Estimating cross-sectional area with the trapezium rule. Question 3(d) — Integration and Area Proving a cubic relation from a shaded area. Question 3(e) — Mean-value Integration Using weighted-average integrals to find a balance point.

NCEA Level 3 · AS91579 2024 Integration 5 questions

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Question 1(b) — Polynomial Algebra Setting up and evaluating a shaded polynomial area. Question 1(e) — Integration and Area Finding area between sec² and tan² curves. Question 2(b) — Definite Integral Equations Solving for a lower limit from a definite integral. Question 2(d) — Trigonometric Integration Using trig identities in a shaded sin squared x area. Question 3(d) — Trigonometric Integration Splitting a cosine area into two equal parts.

NCEA Level 3 · AS91579 2023 Integration 3 questions

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Question 1(c) — Integration and Area Finding intersections and the area between square root of x and x squared divided by 8. Question 1(e) — Integration and Area Combining curve intersections with a subtract-a-cap area strategy. Question 3(c) — Trigonometric Integration Using product-to-sum in a shaded trigonometric area.

NCEA Level 3 · AS91579 2022 Integration 6 questions

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Question 1(b) — Trigonometric Integration Using symmetry and signed area on a graph with no formula. Question 1(d) — Integration and Area Setting up a shaded-area equation to solve for k. Question 1(e) — Integration and Area Finding exact area between exponential curves. Question 2(b) — Definite Integral Equations Solving for an upper limit from a definite integral. Question 3(d) — Integration and Area Identifying top-minus-bottom for an area between a curve and a line. Question 3(e) — Logarithms Expressing a shaded region as rectangle minus logarithmic curve area.

NCEA Level 3 · AS91579 2021 Integration 5 questions

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Question 1(b)(ii) — Integration and Area Using the curve equation to find the area between x = 1 and x = 2. Question 2(b) — Definite Integral Equations Adding a constant inside a definite integral. Question 2(d) — Logarithms Setting a shaded logarithmic area equal to 4 and solving for k. Question 3(d) — Rational-function Integration Finding graph limits and integrating a rational curve. Question 3(e) — Integration and Area Proving an exponential shaded-area formula in terms of k.

NCEA Level 3 · AS91579 2020 Integration 3 questions

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Question 1(d) — Integration and Area Finding intersections and the area between a curve and a line. Question 2(e) — Integration and Area Splitting geometric area where a curve crosses the axis. Question 3(e) — Integration and Area Product-to-sum in an area between cosine curves.

NCEA Level 3 · AS91579 2019 Integration 6 questions

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Question 1(d) — Integration and Area Proving equal areas under a reciprocal-root curve. Question 2(b) — Trigonometric Integration Using signed areas and symmetry on a graph. Question 2(c) — Logarithms Solving for an upper limit from a logarithmic integral. Question 2(d) — Integration and Area Using the cos squared x identity to find shaded area. Question 2(e) — Integration and Area Finding area between two exponential curves. Question 3(e) — Integration and Work Pumping energy with similar triangles and a definite integral.

NCEA Level 3 · AS91579 2018 Integration 5 questions

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Question 1(e) — Trigonometric Integration Using a point on an exponential curve to prove an area result. Question 2(b) — Radicals and Surds Finding an upper limit from a radical integral. Question 3(c) — Definite Integral Equations Solving an exponential definite integral for its limit. Question 3(d) — Integration and Area Intersections and area between a cube-root curve and a parabola. Question 3(e) — Integration and Area Subtracting the tangent area from the area under a quartic.

NCEA Level 3 · AS91579 2017 Integration 5 questions

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Question 1(b) — Integration and Area Using a definite integral to find an enclosed area. Question 1(d) — Definite Integral Equations Solving for an unknown limit in an exponential integral. Question 2(c) — Integration and Area Finding the shaded area between a parabola and its tangent. Question 2(d) — Integration and Area Product-to-sum before integrating a trigonometric area. Question 3(c) — Rational-function Integration Rewriting a rational function to find the exact courtyard area.

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Page updated 24 September 2026.