← Integration methods: past papers

AS 91579 · 2025 paper · trapezium rule, areas, centroid

Question 3(e) Merit to Excellence

Practises Areas under & between curves

Opens with a numerical estimate from measured depths, then a log integral left in terms of a limit, a separable differential equation, an area that must be turned into a cubic, and a balance point found from a ratio of two integrals.

Murray is planning to hang a piece of his art on a wall. Its shape is the region between the x-axis and the curve f(x) = (1/3)(x2 + 3)2, from x = 0 to x = 3. The x-value of the point to hang it from, so that it balances, is given by (int03 x f(x) dx)/(int03 f(x) dx).

Use this formula to find the x-value of the hanging point.

f(x)=(1/3)(x2 + 3)2
0 ≤ x ≤ 3

The 2025 paper, page 13 ↗ Marking schedule, page 6 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 There are two integrals. Start with the top one. What does x f(x) look like?

    x f(x) = (1/3) x(x2 + 3)2. The x outside is half the derivative of x2 + 3, so the reverse chain rule (or u = x2 + 3) will do it.

  2. 2 Integrate it, then put in the limits.

    int03 (1/3) x(x2 + 3)2 dx = Bigl[((x2 + 3)3/18)Bigr]03 = (123 − 33)/18 = 1701/18 = 94.5

  3. 3 The bottom one has no x outside the bracket, so a substitution has nothing to match. What will work instead?

    Expand: (1/3)(x2 + 3)2 = (1/3)(x4 + 6x2 + 9).
    int03 (1/3)(x4 + 6x2 + 9)dx = (1/3)Bigl[x5/5 + 2x3 + 9xBigr]03 = (1/3)(243/5 + 54 + 27) = 216/5 = 43.2

  4. 4 Divide the top by the bottom.

    bar x = 94.5/43.2 = 35/16 = 2.1875

  5. 5 Does that make sense?

    Yes. The art is much taller on the right, so its balance point sits to the right of the middle, x = 1.5.
Answer Hang it at x = 35/16 = 2.1875 (2.19 to 3 s.f.).

What the marker wanted

  • Achieved One of the two integrals correct.
  • Merit Both integrals correct.
  • Excellence The value 2.1875, fully justified, is E8; a correct solution with one minor error is E7.

In our words, from NZQA’s 2025 marking schedule.

✕

Common mistake

Trying a substitution on int (x2 + 3)2 dx, where there is no x outside the bracket for du to match. NZQA’s 2025 report lists not being able to integrate a function that needs rearranging first among the Not Achieved answers.

How did it go?

Got it moves it further out. Review brings it back tomorrow.