2026 Prediction Examination
Level 3 Integration
91579 Apply integration methods in solving problems
Credits: Six
| Achievement | Achievement with Merit | Achievement with Excellence |
|---|---|---|
| Apply integration methods in solving problems. | Apply integration methods, using relational thinking, in solving problems. | Apply integration methods, using extended abstract thinking, in solving problems. |
Instructions
- Suggested time: 60 minutes.
- You should attempt ALL the questions in this paper.
- Show ALL relevant working.
- A calculator is permitted; a graphics calculator is best.
- The formula sheet opens from the margin beside any question.
- Use the margin beside each question for its timer and worked answer.
- Start the timer below when you are ready to begin.
Main Exam Timer
Our 2026 prediction paper, written from the 2021 to 2025 papers and this year's assessment specification. Every question has a worked answer.
A Maths Study Hub practice paper. Not an official NZQA examination.
QUESTION ONE
(a)
Find $\displaystyle\int\left(6\operatorname{cosec}^2(2x)+\frac{4}{\mathrm{e}^{2x}}\right)\mathrm{d}x$.
(b)
On a clear summer day, the solar panels on a house generate electricity at a rate of
$$R=4\sin\left(\frac{\pi t}{12}\right)$$kilowatt-hours (kWh) per hour, where $t$ is the time in hours after 6 am, for $0\le t\le12$.
Find the total amount of electricity the panels generate between 6 am and 6 pm.
You must use calculus and show the results of any integration needed to solve the problem.
(c)
The graph below shows the curves $y=\mathrm{e}^{x}$ and $y=\mathrm{e}^{2-x}$.
Find the area of the shaded region, which is bounded by the two curves and the $y$-axis.
You must use calculus and show the results of any integration needed to solve the problem.
(d)
Consider the differential equation
$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x^2+1}{xy},\quad x\gt 0.$$Given that $y=2$ when $x=1$, find the value of $y$ when $x=\mathrm{e}$.
You must use calculus and show the results of any integration needed to solve the problem.
(e)
The graphs of $y=\sin x$ and $y=k\cos x$, where $k$ is a positive constant, are shown below for $0\le x\le\frac{\pi}{2}$. They meet at the point P.
The shaded region lies under both curves: it is bounded by the two curves and the $x$-axis.
Find the value of $k$ for which the shaded region has an area of $\frac12$.
You must use calculus and show the results of any integration needed to solve the problem.
QUESTION TWO
(a)
Find $\displaystyle\int\frac{(x+3)^2}{\sqrt x}\,\mathrm{d}x$.
(b)
Find the value of $k$, given that
$$\int_0^{\ln k}4\mathrm{e}^{2x}\,\mathrm{d}x=48.$$You must use calculus and show the results of any integration needed to solve the problem.
(c)
The graph below shows the curve $y=\dfrac{4x+1}{x+2}$ and the line $y=4$, for $x\ge0$.
The shaded region between the curve and the line, from $x=0$ to $x=k$, has an area of 7 units².
Find the value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.
(d)
A car is travelling at $20\text{ m s}^{-1}$ when the driver starts braking. While the car slows down, its acceleration is modelled by
$$a(t)=-\frac{6}{\sqrt{t+1}}$$where $a(t)$ is the acceleration in m s−2, and $t$ is the time in seconds since the brakes were first applied.
How far does the car travel from when the brakes are first applied until it stops?
You must use calculus and show the results of any integration needed to solve the problem.
(e)
Ruby wants to check the temperature inside her café’s freezer, so she puts a thermometer in it. When she puts it in, the thermometer reads 14 °C. One minute later it reads −2 °C, and after another minute it reads −10 °C.
The rate at which the thermometer’s reading changes is proportional to the difference between the reading and the temperature inside the freezer, which stays constant.
Write a differential equation that models this situation, and solve it to find the temperature inside the freezer.
You must use calculus and show the results of any integration needed to solve the problem.
QUESTION THREE
(a)
Hannah is making a template for a new surfboard. The template is 180 cm long. She measures its width every 30 cm along its length, as shown in the diagram.
Use Simpson’s rule and these measurements to find an approximation for the area of the template.
(b)
Solve the differential equation
$$\frac{\mathrm{d}y}{\mathrm{d}x}=6x\sqrt{x^2+5},$$given that $y=10$ when $x=2$.
You must use calculus and show the results of any integration needed to solve the problem.
(c)
The graph below shows the function $y=2\sin(3x)\cos x$ for $0\le x\le\frac{\pi}{2}$. It crosses the $x$-axis at $x=\frac{\pi}{3}$.
Find the total area of the shaded regions.
You must use calculus and show the results of any integration needed to solve the problem.
(d)
A curve has second derivative
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=4\cos^2x.$$The curve has a stationary point at $(\pi,1)$.
Find the value of $y$ when $x=\frac{\pi}{2}$.
You must use calculus and show the results of any integration needed to solve the problem.
(e)
A floating pontoon on a lake is 6 m long. Every cross-section across it is the same shape: the region between the curve
$$y=\mathrm{e}^{x}+\mathrm{e}^{-x}-2$$and the line $y=\frac43$, which is the pontoon’s deck. Here $x$ and $y$ are in metres. The diagrams show the pontoon, and its cross-section.
A floating object sinks until the water it pushes aside has the same mass as the object and its load. The water it pushes aside has the same volume as the part of the object that is under the water. 1 m³ of the lake’s water has a mass of 1000 kg.
The pontoon has a mass of 4000 kg. Find the greatest mass of people and equipment that it can carry before its deck is level with the water.
You must use calculus and show the results of any integration needed to solve the problem.
How we chose these questions
A prediction paper is our best judgement of what the 2026 examination is likely to ask. It isn’t a copy of any past paper, and nobody can know the real questions in advance. We built this one from NZQA’s 2021 to 2025 Integration examinations and marking schedules, the examiners’ reports on those years, and the 2026 assessment specification, and we checked every paper back to 2013 so that no question repeats one.
Every question is here for at least one of three reasons
- It’s in the 2026 specificationThe specification says candidates must show the results of any integration, may have to form their own equations for Merit and Excellence, and may need to manipulate exponents and logarithms to solve differential equations. Excellence questions may ask candidates to devise their own model.
- It comes up every yearEvery paper since 2013 has had a separable differential equation and areas under or between curves, and nearly every one has had a numerical method and a kinematics problem. Since 2020 every paper has had a fraction to divide before integrating, and most recent papers have used the product-to-sum formulas. All of them are here.
- It’s where marks are lostThe examiners’ reports name the mistakes candidates make most often: leaving out the constant of integration or assuming it is zero, losing a factor when integrating, turning every fraction into a logarithm, mistakes with square roots as powers, and the wrong strip width in Simpson’s rule. We built these into the questions.
Question by question
| Question | What it tests | Why it’s here |
|---|---|---|
| 1(a) A | Integrating a reciprocal trigonometric function and an exponential | The reports from 2022 to 2025 say candidates often lose a factor when integrating; here there is a factor to divide by in each term, and a fraction that is not a logarithm. |
| 1(b) A | The electricity solar panels generate in a day, from its rate | Rates of change are in the standard. This needs the factor from a trigonometric function and a calculator in radians. |
| 1(c) M | The area between two exponential curves | An area between curves is in most papers. The 2024 report praises candidates who manage negatives when integrating, which this needs. |
| 1(d) M | Solving a differential equation by separating the variables | A separable equation is in every paper. The 2021 report warned about integrating y wrongly and about constants that change when an equation is doubled. |
| 1(e) E | The area under two trigonometric curves that meet at an unknown point | The 2023 report says Excellence candidates could split an area into parts, and the 2022 report that they could work with exact values. This needs both. |
| 2(a) A | Expanding and dividing by a square root before integrating | The 2025 report lists not writing square roots as powers as a reason for Not Achieved. |
| 2(b) A | Finding an unknown limit, where the limit is a logarithm | An unknown limit is in most papers, and the 2024 report credits candidates who solve the equation with logarithms that follows. |
| 2(c) M | The area between a curve and the line it approaches | A fraction to divide before integrating is in every paper since 2020, and the reports from 2023 to 2025 say candidates turn every fraction into a logarithm. |
| 2(d) M | How far a braking car travels before it stops | Kinematics is in nearly every paper. The 2025 report says many candidates assumed the constant was zero because the displacement started at zero; here it isn’t. |
| 2(e) E | Finding a freezer’s temperature from a thermometer that hasn’t settled | A differential equation in a context was in every Excellence set from 2021 to 2024, and Newton’s Law of Cooling is named in the standard. Here the surrounding temperature is the unknown. |
| 3(a) A | Simpson’s rule: the area of a surfboard template | A numerical method is in nearly every paper, and the trapezium rule was used in 2025. The 2023 and 2025 reports say candidates use the wrong strip width. |
| 3(b) A | A differential equation solved with the reverse chain rule | The reverse chain rule is in the standard, and every paper has a differential equation with a starting condition. |
| 3(c) M | A product of trigonometric functions, with part of the area below the axis | The product-to-sum formulas were in four of the last five papers. The 2023 report says Merit candidates recognised that areas below the axis count as negative. |
| 3(d) M | Integrating twice, using a stationary point | The standard includes equations of the form y″ = f(x). The 2022 report said integrating a squared trigonometric function without an identity was common. |
| 3(e) E | How much a floating pontoon can carry, from the area of its cross-section | NZQA set a real-life context with a rule from outside the standard at Excellence in 2024 and 2025. Here the rule is how things float, and the integration is an area between a curve and a line, whose limits come from a quadratic in eˣ, as in 2022. |
None of the Excellence questions, 1(e), 2(e) and 3(e), repeats an NZQA question from 2013–2025.
NZQA sources: 2026 assessment specification, 2025 assessment report, 2025 examination, 2025 assessment schedule. The 2021 to 2024 papers, schedules and reports are on the same NZQA pages.
Maths Study Hub is independent and is not affiliated with or endorsed by NZQA. Written October 2026.