Free practice / JC1 H2 Maths Promo

Applications of integration: NJC 2024 Promo Q3

Attempt the question on paper, then open the worked solution to check your method.

National Junior College2024 PromoPromo Q3●○○ Foundational5 marks

What this question tests

Areas and volumes by integration.

Question

Given that $k$ is a positive constant, sketch the curve $E$ with equation $y = \left|x^2(x - k)\right|$ , labelling the coordinates of the points where the curve cuts the axes. [2]
Hence, find, in terms of $k$ , the area of the region(s) bounded by $E$ and the $x$ -axis for $0 \leq x \leq 2k$ . [3]

Show full worked solution▾

(Applications of Integration, Transformation)

Suggested Solution

(i)

$National Junior College 2024 Promo Q3: solution diagram$

(ii) $\begin{aligned} \int_0^{2k}|x^2(x-k)|\,\mathrm{d} x &=\int_0^{2k}|x^3-kx^2|\,\mathrm{d} x\\ &=-\int_0^k(x^3-kx^2)\,\mathrm{d} x+\int_k^{2k}(x^3-kx^2)\,\mathrm{d} x\\ &=-\left[\frac{x^4}{4}-k\frac{x^3}{3}\right]_0^k +\left[\frac{x^4}{4}-k\frac{x^3}{3}\right]_k^{2k}\\ &=\frac32k^4. \end{aligned}$

Answers

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

← All Applications of integration questions JC H2 Maths tuition at Math Academy