MathsGradeUp logoMathsGradeUp
MathematicsGCSE Higher & A Level

What is area under a curve?

The area under a curve is the region between the graph and the x-axis, found exactly by definite integration or estimated using the trapezium rule.

Area under a curve explained

At GCSE Higher you estimate the area by splitting it into trapezia and summing them; at A Level you integrate for the exact value. The context matters: under a speed-time graph the area is distance travelled.
For the area between two curves, integrate the difference of the functions between their points of intersection.

Key formula

Area=∫ab(yupper−ylower)dx\text{Area} = \displaystyle\int_{a}^{b} \left(y_{\text{upper}} - y_{\text{lower}}\right)dx

Worked example

The area between y=x+2y = x + 2 and y=x2y = x^{2} runs from x=−1x = -1 to x=2x = 2 and equals ∫−12(x+2−x2)dx=4.5\displaystyle\int_{-1}^{2}\left(x + 2 - x^{2}\right)dx = 4.5.

Examiner tip

Always sketch first. Deciding which curve is on top is worth more marks than the integration itself.

Area under a curve: common questions

Is the trapezium rule an overestimate or underestimate?
It overestimates for a curve that bends upwards (convex) and underestimates for one that bends downwards, because the chords sit above or below the curve.
How do I find the area between two curves?
Solve the equations simultaneously for the limits, then integrate upper curve minus lower curve between those limits.

Related terms

Definite integral

A definite integral evaluates an integral between two limits and returns a single numerical value, usually representing signed area.

Integration

Integration is the reverse of differentiation: it recovers a function from its rate of change and measures the area under a curve.

Trapezium rule

The trapezium rule estimates the area under a curve by dividing it into strips of equal width and treating each strip as a trapezium.

Practise mathematics on MathsGradeUp

Take area under a curve from a definition to full marks with exam-board question banks, revision notes and instant step-by-step feedback from Study Coach.

Maths revision notesPractice questionsCreate a free account

Related on MathsGradeUp

Keep studying

Practise it