Question 2
Determine the approximate area $\left(\text{units}^2\right)$ between the curve and the $x$-axis for $0\le x\le8$ by calculating the area of the shaded rectangles.
Not quite. 15 is the sum of the heights, $0+2+4+9$. Each rectangle is 2 units wide, so each area is $2\times$ its height. In 2025, 2.0% of Queensland students chose (A).
Not quite. 28 is what you get if you read the tallest rectangle as 8. Its top sits on the 9 gridline, so it adds $2\times9=18$, not 16. In 2025, 6.2% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2025, 89.4% of Queensland students chose (C).
Not quite. 31 is one square unit more than the shaded rectangles cover. Check how you read each height and width off the grid. Every top sits exactly on a gridline, at 2, 4 and 9, and each rectangle is 2 wide: $4+8+18=30$. In 2025, 2.1% of Queensland students chose (D).
The first strip runs from 0 to 2. The curve is at 0 on its left edge, so there’s no rectangle here. It adds $2\times0=0$.
Each strip is 2 units wide, and its height is the curve’s value at the left edge. So each rectangle’s area is $2\times$ that height. Add the next strip.
All four strips are in. The shaded rectangles cover $0+4+8+18=30$ square units. Every rectangle sits under the curve, so this underestimates the true area.
Read each shaded rectangle off the grid. Every one is 2 units wide.
| Strip | $[2,4]$ | $[4,6]$ | $[6,8]$ |
|---|---|---|---|
| Height | 2 | 4 | 9 |
There’s no rectangle on $[0,2]$ because the curve is at 0 on that strip’s left edge.
Area of a rectangle is width times height.
$$\begin{aligned}2\times2&=4\\2\times4&=8\\2\times9&=18\end{aligned}$$
You can check by counting grid squares: 4, 8 and 18 of them.
Add the three areas.
$$4+8+18=30\ \text{units}^2$$
This is an estimate, not the exact area. The curve sits above every rectangle, so the true area is larger.
$$\begin{aligned}A&\approx2\times2+2\times4+2\times9\\&=4+8+18\\&=30\ \text{units}^2\end{aligned}$$
The strip from 0 to 2 has no shaded rectangle, so it adds nothing.
How Queensland students answered in 2025
From the QCAA Mathematical Methods subject report for 2025. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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