13.5PureStretch

Areas Under Curves

The big payoff of integration: a definite integral gives the exact area between a curve and the x-axis. Set the limits to the x-values bounding the region and evaluate.

25 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.5 Areas under CurvesWatch the full walkthrough before the notes below.

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What you'll be able to do

Use a definite integral to find an area under a curve
Choose the correct limits
Find limits from where the curve meets the x-axis
Evaluate the resulting area

Area as a definite integral

The area between $y = f (x)$ and the $x$ -axis, from $x = a$ to $x = b$ (with the curve above the axis), is the definite integral.

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

For a region above the

x

-axis between

x = a

and

x = b

\int_{0}^{3} x^{2} d x = [\frac{x ^{3}}{3}]_{0}^{3}

= \frac{27}{3} - 0

Answer

9

Finding the limits

Often the region is bounded by where the curve crosses the $x$ -axis. Find those roots first ( $y = 0$ ) and use them as the limits.

1Roots:

x = 0

and

x = 4

— these are the limits.

2Integrate

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

= 32 - \frac{64}{3}

Answer

\frac{32}{3}

Tip — If limits are not given, they are usually the curve’s x-intercepts — solve y = 0 to find them.

Formula recap

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

Area under a curve (above the axis).

y = 0 \Rightarrow limits a, b

Limits from x-intercepts.

Common mistakes to avoid

Using arbitrary limits instead of the region’s boundaries.

Use the given limits, or the x-intercepts of the bounded region.

Forgetting to evaluate (leaving it as an indefinite integral).

An area is a number — substitute the limits and subtract.

Key takeaways

Area under a curve (above the axis) = ∫ from a to b of y dx.
If not given, the limits are usually the x-intercepts (solve y = 0).
Evaluate the definite integral to get the numerical area.

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