Area Between Curves
Integrate the difference of two functions.
Key idea
Integrate the difference of two functions.
In depth
Area = ∫ₐᵇ (top − bottom) dx. Find intersection points for the limits and ensure you subtract the lower curve from the upper one. If curves swap order, split the integral.
Worked example
Between y = x and y = x² on [0, 1]: ∫(x − x²) dx = 1/2 − 1/3 = 1/6.
Formula
Real-world uses
- Finding the area between a supply and demand curve in economics or two road profiles
Common mistake
Using the wrong order of curves, giving negative area.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Area Between Curves?
Integrate the difference of two functions. It is a topic at the advanced level in Calculus.
What is the formula for Area Between Curves?
The key formula is A = ∫ (f−g) dx.
Where is Area Between Curves used in real life?
Real-world examples: Finding the area between a supply and demand curve in economics or two road profiles.
What should I learn before Area Between Curves?
Make sure you are comfortable with Improper Integrals, the previous topic in Calculus.
What is a common mistake with Area Between Curves?
Using the wrong order of curves, giving negative area.
How can I practise Area Between Curves?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Area Between Curves?
Move on to Volumes by Disks and Washers next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.