13.7PureStretch

Areas Between Curves and Lines

To find the area trapped between two graphs, integrate the difference “top minus bottom”. The limits are the x-values where the graphs intersect — so you usually solve them simultaneously first.

30 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.7 Areas between Curves and LinesWatch the full walkthrough before the notes below.

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What you'll be able to do

Find the area between a curve and a line
Use “top function minus bottom function”
Find limits from the intersection points
Combine integration with simultaneous equations

Top minus bottom

The area between two graphs is the integral of the $upper$ function minus the $lower$ one, between their intersection points.

Area = \int_{a}^{b} (y_{top} - y_{bottom}) d x

Subtract the lower curve from the upper, then integrate.

Find the limits first

The limits $a$ and $b$ are where the two graphs cross. Set the functions equal and solve to find these intersection $x$ -values.

1Set equal:

x^{2} = x + 2 \Rightarrow x^{2} - x - 2 = 0

(x - 2) (x + 1) = 0

Answerlimits

x = - 1

and

x = 2

Tip — Decide which graph is on top within the region (test a value) — that is your “top” function.

Putting it together

Integrate the difference between the limits. Subtracting first means you do not have to worry about parts below the $x$ -axis separately — the difference handles it.

\int_{- 1}^{2} ((x + 2) - x^{2}) d x = [\frac{x ^{2}}{2} + 2 x - \frac{x ^{3}}{3}]_{- 1}^{2}

2Evaluate top minus bottom limit.

Answer

\frac{9}{2}

Formula recap

Area = \int_{a}^{b} (y_{top} - y_{bottom}) d x

Top minus bottom.

y_{1} = y_{2} \Rightarrow intersection limits

Solve for the limits.

Common mistakes to avoid

Subtracting bottom minus top (giving a negative area).

Always do upper function minus lower function.

Guessing the limits instead of finding the intersections.

Set the two equations equal and solve for the limits.

Key takeaways

Area between graphs = ∫ (top − bottom) dx.
Limits are the intersection points (solve the equations simultaneously).
Subtracting first deals with regions below the axis automatically.

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