Worked example 1
Find an enclosed area
Find the area enclosed by y = 2x + 3 and y = x².
- Solve x² = 2x + 3 to get x = −1 and x = 3.
- The line is above the parabola between the roots.
- Integrate (2x + 3 − x²) from −1 to 3
Answer: square units
Higher Mathematics
Higher Maths Area between curves
Practise area between curves for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Find intersections and integrate top function minus bottom function between the correct limits.
Before you start
- Find intersections.
- Evaluate definite integrals.
- Decide which graph is above another.
Explanation
The area between two curves is the integral of top minus bottom between their intersection points.
If the curves swap order, split the integral so each piece contributes positive area.
Visual support
Area = ∫ from 0 to 2 of [(4x − x²) − x²] dx
Method and rules
- Area = integral from a to b of [top(x) − bottom(x)] dx
- Find a and b by solving f(x) = g(x).
- Split at any point where the upper curve changes.
Worked examples
Worked example 2
Method check
What integrand gives area when f is above g?
- Use top minus bottom.
So: f(x) − g(x).
Watch out
- Integrating before finding the intersection limits.
- Subtracting top from bottom in the wrong order.
- Reporting a negative value as an area.
Exam reminder
For area between curves, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
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