Find the area enclosed by y=x+4 and y=x³+3x²+x.

01 / Standard#Your turn

Official paper · jm02-2023 · 2(b)(ii) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find all intersection abscissae.

Hint 2

Subtract the lower curve from the upper line.

Worked solution

Factor the difference.
$x^3+3x^2-4=(x-1)(x+2)^2=0\implies x=-2,1$
The integrand is nonnegative on the bounded interval.
$A=\int_{-2}^1(4-x^3-3x^2)\,dx=\left[4x-\frac{x^4}4-x^3\right]_{-2}^1=\frac{27}4$

Area 27/4.

Checks and common pitfalls: Tangency gives a repeated intersection root but does not eliminate the bounded region.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Factor the difference.
$x^3+3x^2-4=(x-1)(x+2)^2=0\implies x=-2,1$
The integrand is nonnegative on the bounded interval.
$A=\int_{-2}^1(4-x^3-3x^2)\,dx=\left[4x-\frac{x^4}4-x^3\right]_{-2}^1=\frac{27}4$

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

2023 JM02