Find the area between the two parabolas.

01 / Standard#Your turn

y=x^2-8x+24,\quad y=8x-x^2

Official paper · jm02-2022 · 2(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Equate the two functions for the limits.

Hint 2

The downward parabola is the upper curve between intersections.

Worked solution

Find x=2 and x=6.
$2x^2-16x+24=2(x-2)(x-6)=0$
Integrate upper minus lower.
$A=\int_2^6(-2x^2+16x-24)\,dx=\left[-\frac23x^3+8x^2-24x\right]_2^6=\frac{64}3$

Area 64/3.

Checks and common pitfalls: Signed integrals must be arranged with a nonnegative height for geometric area.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Find x=2 and x=6.
$2x^2-16x+24=2(x-2)(x-6)=0$
Integrate upper minus lower.
$A=\int_2^6(-2x^2+16x-24)\,dx=\left[-\frac23x^3+8x^2-24x\right]_2^6=\frac{64}3$

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

2022 JM02