Question 1(c) Merit

A linear-plus-reciprocal integral, a velocity in km hr⁻¹ integrated to a distance from a point 3 km away, an area between a root and a parabola, a separable equation, and an area bounded by three different curves at once.

The graph shows y = √x and 8y = x², with the region between them shaded, from the origin to the point where they cross again.

Find the shaded area enclosed between y = √x and 8y = x².

The 2023 paper, page 3 ↗ Marking schedule, page 1 ↗

Work it out

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

1. 1 Where do the curves meet? Write the second one as y = …

   8y = x² is y = x²/8. They meet where √x = x²/8: squaring, x = x⁴/64, so x(x³ − 64) = 0, and x = 0 or x = 4.

2. 2 Which is on top between 0 and 4?

   √x. Try x = 1: √1 = 1 but 1/8 = 0.125.
   Area = int₀⁴ (x^1/2 − x²/8)dx

3. 3 Integrate and put in the limits.

   Bigl[(2/3) x^3/2 − (x³/24)Bigr]₀⁴ = (2/3)(8) − 64/24 = 16/3 − 8/3 = 8/3

What the marker wanted

• Achieved The integral correct.
• Merit The area, 8/3, from correct integration.

In our words, from NZQA’s 2023 marking schedule.

How did it go?

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