← Integration methods: past papers

AS 91579 · 2024 paper · expansion, constants, DE, mass

Question 2(b) Achieved

Practises Standard integrals

Expand-then-integrate, a lower limit recovered from a definite integral, a separable equation in y², an area between a curve and a horizontal line, and a physical integral left entirely in symbols.

Find the value of k, given that intk16 3√x dx = 112.

The 2024 paper, page 6 ↗ Marking schedule, page 3 ↗

Work it out

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

  1. 1 Write 3√x as a power and integrate.

    int 3x1/2 dx = 3x3/2/(3/2) = 2x3/2

  2. 2 Put in the limits k and 16, and set it equal to 112.

    2(16)3/2 − 2k3/2 = 112
    163/2 = 64, so 128 − 2k3/2 = 112 and k3/2 = 8.

  3. 3 Undo the power 3/2.

    k = 82/3 = 4
Answer k = 4

What the marker wanted

  • Achieved k = 4, with correct integration shown.

In our words, from NZQA’s 2024 marking schedule.

✕

Common mistake

Integrating √x without writing it as x1/2 first. NZQA’s 2024 report lists not rewriting a root as a power among the Not Achieved answers.

How did it go?

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