Areas | Year 12 | Ascent Library

01 / LEARN

The idea, step by step

Set the region up first

A region between curves y=f(x) and y=g(x) has area $\int_{a}^{b} (upper - lower) d x$ . First find the intersections and identify which curve is above the other. A small test value between intersections helps.

If the order switches, split at the crossing. For area between a curve and the axis, split at its zeros and make each subarea non-negative. Drawing the boundary prevents accidental inclusion of an unwanted region.

Respect the chosen variable

Horizontal slices integrate right boundary minus left boundary with respect to y. This can simplify a region whose curves are awkward to solve or whose vertical slices require several pieces.

Area has square units. A signed integral can cancel while the region has positive area, so an exact zero is a reason to revisit the geometry rather than automatically report no region.

A common trap

Take the absolute value of the final signed integral to get every total area.

Instead: Cancellation may already have occurred. Split into non-negative subareas before adding.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find the area between y=x and y=x squared from x=0 to 1.

On this interval x is the upper function.
Integrate x-x squared to obtain 1/2-1/3.

1 / 6

square units

Example 2

Worked example 2

Find total area between y=x squared-1 and the x axis from -2 to 2.

Split at -1 and 1, using symmetry.
Twice [integral from 0 to 1 of (1-x squared) plus integral from 1 to 2 of (x squared-1)] equals 4.

4

square units

Check your understanding

What are the intersection x values for x squared=2x?

Show answer

0 and 2.

Factor x(x-2)=0.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the area between y=x squared and y=2x.
2 marks
Find the area under y=4-x squared above the x axis.
2 marks
Find the area between y=|x| and the axis on [-1,1].
2 marks
Find the area between y=x and y=sin x on [0,pi].
2 marks
Find area between x=y squared and x=4 for 0<=y<=2.
2 marks
A student reports area zero between y=x cubed and the x axis on [-1,1]. Correct the result.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the area between y=x squared and y=2x.
2 marks
The intersections are 0 and 2.
Integrate 2x-x squared over that interval.
$4 / 3$ square units
Find the area under y=4-x squared above the x axis.
2 marks
Zeros are -2 and 2.
Integrate 4-x squared between them.
$32 / 3$ square units
Find the area between y=|x| and the axis on [-1,1].
2 marks
Use two congruent right triangles.
Each has area 1/2.
$1$ square unit
Find the area between y=x and y=sin x on [0,pi].
2 marks
x is at least sin x throughout this interval.
Integrate x-sin x and evaluate endpoints.
$π^{2} / 2 - 2$ square units
Find area between x=y squared and x=4 for 0<=y<=2.
2 marks
Horizontal slices have width 4-y squared.
Integrate from y=0 to 2.
$16 / 3$ square units
A student reports area zero between y=x cubed and the x axis on [-1,1]. Correct the result.
2 marks
The signed integral is zero, but areas add.
Use twice the integral of x cubed from 0 to 1.
$1 / 2$ square unit

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics integration techniques and applications, including inverse-trigonometric patterns, substitution, parts, simple partial fractions, areas and solids of revolution.

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