Area Under a Curve
Master area under a curve for GCSE Maths with structured, exam-style practice. This Higher resource covers estimating the area under a curve and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Split the area into trapeziums or rectangles to estimate it.
Free downloads
These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
The area under a curve represents the total accumulated quantity. On a velocity-time graph, the area gives the distance travelled, because velocity multiplied by time is distance.
Since a curve does not form neat rectangles, the area is estimated. The standard method divides the region into strips — usually trapeziums — and adds their areas together.
The estimate improves as the strips get narrower. Whether it over- or underestimates depends on the curve's shape: for a curve bending upwards, straight-topped trapeziums sit above the curve and overestimate the area.
Revision notes
Using trapeziums
Divide the region into vertical strips of equal width and treat each as a trapezium.
The area of each is \(\frac{1}{2}(a + b)h\), where \(a\) and \(b\) are the parallel vertical sides and \(h\) is the strip width. Add all the strips for the total.
Interpreting the area
On a velocity-time graph, area gives distance. On other graphs it gives whatever the product of the two axes represents.
Always state units, which come from multiplying the units of the two axes, such as metres per second times seconds giving metres.
Over and underestimates
If the curve bends upwards, the straight tops of the trapeziums lie above it, so the estimate is too large.
If the curve bends downwards, the strips fall below and the estimate is too small. Questions often ask you to say which and why.
Key points
- Area under a curve gives an accumulated total.
- On a velocity-time graph it gives distance.
- Divide into strips and treat each as a trapezium.
- Trapezium area is \(\frac{1}{2}(a+b)h\).
- Narrower strips give a better estimate.
- The shape of the curve decides over or underestimate.
Worked examples
Example 1
Find the area of one trapezium strip with parallel sides \(4\) and \(6\) and width \(2\).
Working
Example 2
Two strips have areas \(10\) and \(14\). Estimate the total area.
Working
Example 3
A velocity-time curve bends upwards. Is a trapezium estimate too large or too small?
Working
Common mistakes
Forgetting to halve in the trapezium formula.
The area is ½(a + b)h, so the sum of the parallel sides is halved.
Using the wrong strip width.
The width h is the horizontal distance, not one of the vertical sides.
Omitting units.
Units come from multiplying the axis units, such as m/s × s = m.
Guessing whether it is an over or underestimate.
Look at which way the curve bends relative to the straight strip tops.
Exam tips
- Draw the strips on the graph before calculating.
- Use equal strip widths so the arithmetic stays simple.
- Work out each strip separately, then add.
- State clearly whether the estimate is too large or too small, and why.
Key terms
- Trapezium
- A quadrilateral with one pair of parallel sides.
- Estimate
- An approximate value.
- Overestimate
- A value larger than the true one.
- Velocity-time graph
- A graph whose area gives distance travelled.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.