11.8PureStretch

Finding Areas

A definite integral gives the signed area between a curve and the x-axis. To find a true area you must handle regions below the axis and areas between two curves carefully.

26 min Video by Zeeshan Zamurred Integration

Edexcel A level a Maths: 11.8 Finding Areas (Part 1)Watch the full walkthrough before the notes below.

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What you'll be able to do

Find the area under a curve by definite integration
Handle areas below the x-axis
Find the area between two curves
Combine regions correctly

Area under a curve

The area between $y = f (x)$ , the $x$ -axis and $x = a, x = b$ is $\int_{a}^{b} f (x) d x$ — provided the curve is above the axis throughout.

Area = \int_{a}^{b} f (x) d x

Area under a curve (above the axis).

Below the axis and between curves

Where the curve is below the axis the integral is negative, so split the region and take the modulus of each part. The area between two curves is $\int_{a}^{b} (upper - lower) d x$ .

1Upper is

y = x

\int_{0}^{1} (x - x^{2}) d x

= [\frac{x ^{2}}{2} - \frac{x ^{3}}{3}]_{0}^{1} = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}

Answer

\frac{1}{6}

Tip — A negative integral means the region is below the axis — take its modulus for the area.

Formula recap

Area = \int_{a}^{b} f (x) d x

Above the axis.

\int_{a}^{b} (upper - lower) d x

Between two curves.

Common mistakes to avoid

Treating a negative integral as a negative area.

Area is positive; take the modulus of the below-axis part.

Subtracting lower from upper the wrong way.

Always upper minus lower for area between curves.

Key takeaways

Area under a curve = ∫ₐᵇ f(x) dx (above axis).
Below the axis: take the modulus of that part.
Between curves: ∫(upper − lower) dx.

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