Find the area enclosed by the two cubics.

01 / Standard#Your turn

y_1=x^3+3x^2-4,\quad y_2=x^3-3x+2

Official paper · jm02-2025 · 2(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Subtract the curves; their cubic terms cancel.

Hint 2

Solve for the two intersection abscissas.

Worked solution

Find where their difference vanishes and its sign between the roots.
$y_2-y_1=-3(x+2)(x-1)>0\quad(-2<x<1)$
Integrate the positive difference.
$A=\int_{-2}^1(-3x^2-3x+6)\,dx=\left[-x^3-\frac32x^2+6x\right]_{-2}^1=\frac{27}2$

Area 27/2.

Checks and common pitfalls: Subtracting the curves before integration simplifies the calculation.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Find where their difference vanishes and its sign between the roots.
$y_2-y_1=-3(x+2)(x-1)>0\quad(-2<x<1)$
Integrate the positive difference.
$A=\int_{-2}^1(-3x^2-3x+6)\,dx=\left[-x^3-\frac32x^2+6x\right]_{-2}^1=\frac{27}2$

Think first. Reveal a hint when the class is ready.

2025 JM02