Area Between Curves
Integrate top minus bottom to find the area between graphs.
About this topic
The area between two curves f(x) on top and g(x) below, from x = a to x = b, is ∫ₐᵇ (f(x) − g(x)) dx. Find the intersection points to get the limits, decide which curve is on top, and integrate the difference. If the curves cross, split the integral at each crossing.
The idea is a Riemann sum of thin vertical strips, each with height top minus bottom. Subtracting automatically handles regions that dip below the x-axis, since both curves shift together. Sometimes it is easier to integrate with respect to y, using right minus left, when curves are given as x in terms of y.
Example: find the area enclosed by y = x and y = x². They meet where x = x², at x = 0 and x = 1. On that interval x is above x², so the area is the integral of (x − x²) dx from 0 to 1, which equals 1/2 − 1/3 = 1/6.
Applications include comparing revenue and cost curves to find total profit, and finding the cross-sectional area of a river bed or an aircraft wing. Tips: sketch the graph first, test a point to see which curve is on top, and remember area is never negative.
Key ideas
- Find intersections first
- Subtract lower from upper
- Split if the curves cross
Quick check
Use cases: where Area Between Curves helps
- Homework: review the key idea behind Area Between Curves before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Area Between Curves without notes.
- Visual learning: play with the interactive lab to see how Area Between Curves changes when you adjust its inputs.
- Teaching a friend: explain Area Between Curves in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Area Between Curves?
Integrate top minus bottom to find the area between graphs.
What will I learn in Area Between Curves?
Find intersections first. Subtract lower from upper. Split if the curves cross.
Is Area Between Curves part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Area Between Curves lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Area Between Curves?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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