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Applications of integration: NJC 2024 Promo Q3

National Junior College2024 PromoPromo Q3●○○ Foundational5 marks

What this question tests

Areas and volumes by integration.

Question

  1. Given that kk is a positive constant, sketch the curve EE with equation y=∣x2(x−k)∣y = \left|x^2(x - k)\right|, labelling the coordinates of the points where the curve cuts the axes. [2]

  2. Hence, find, in terms of kk, the area of the region(s) bounded by EE and the xx-axis for 0≤x≤2k0 \leq x \leq 2k. [3]

Show full worked solution▾

(Applications of Integration, Transformation)

Suggested Solution

(i)

National Junior College 2024 Promo Q3: solution diagram

(ii) ∫02k∣x2(x−k)∣ dx=∫02k∣x3−kx2∣ dx=−∫0k(x3−kx2) dx+∫k2k(x3−kx2) dx=−[x44−kx33]0k+[x44−kx33]k2k=32k4.\begin{aligned} \int_0^{2k}|x^2(x-k)|\,\mathrm{d} x &=\int_0^{2k}|x^3-kx^2|\,\mathrm{d} x\\ &=-\int_0^k(x^3-kx^2)\,\mathrm{d} x+\int_k^{2k}(x^3-kx^2)\,\mathrm{d} x\\ &=-\left[\frac{x^4}{4}-k\frac{x^3}{3}\right]_0^k +\left[\frac{x^4}{4}-k\frac{x^3}{3}\right]_k^{2k}\\ &=\frac32k^4. \end{aligned}

Answers

(ii)Area =32k4= \dfrac{3}{2}k^4

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