13.6PureStretch

Areas Under the x-axis

When a curve dips below the x-axis, the integral there comes out negative. Area itself is always positive, so you must handle the below-axis parts carefully — and never just integrate straight across a root.

25 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.6 Areas Under The x-axisWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand why integrals can be negative
Find the true area when a curve goes below the axis
Split an integral at the roots
Combine areas as positive values

Negative integrals

For a region $below$ the $x$ -axis, the definite integral is negative — it is the signed area. The actual area is its magnitude (drop the minus sign).

area below axis = \int_{a}^{b} y d x

Take the modulus for the physical area.

Split at the roots

If a curve crosses the axis within the region, the positive and negative parts would cancel if integrated together. So $split$ the integral at each root, evaluate each piece, take the modulus of any negative ones, then add.

1Integrate from 0 to 2 and from 2 to 4 separately.

2Take the modulus of whichever piece is negative.

3Add the two positive areas.

Answertotal area = |part 1| + |part 2|

Tip — Never integrate straight across a root for an area — the parts cancel and you get the wrong (too small) answer.

A worked value

A single below-axis region just needs the modulus.

\int_{0}^{2} (x^{2} - 4) d x = [\frac{x ^{3}}{3} - 4 x]_{0}^{2} = \frac{8}{3} - 8 = - \frac{16}{3}

2Area is the modulus.

Answer

\frac{16}{3}

Formula recap

area = \int_{a}^{b} y d x

Below-axis area is positive.

split at each root, then add moduli

When the curve crosses the axis.

Common mistakes to avoid

Integrating across a root and reporting the (cancelled) result as the area.

Split at the root, take moduli, then add.

Reporting a negative number as an area.

Area is always positive — take the modulus.

Key takeaways

Below the axis, the integral is negative (signed area).
Physical area = the modulus of the integral.
If the curve crosses the axis, split at the roots and add the moduli.

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