Area Between Curves
Formula for calculating the area between two curves using definite integrals, with vertical and horizontal methods.
The Formula
Where f(x) is the upper curve and g(x) is the lower curve between x = a and x = b.
Variables
| Symbol | Meaning |
|---|---|
| A | The area between the two curves |
| f(x) | The upper function (larger y-values) |
| g(x) | The lower function (smaller y-values) |
| a, b | The x-values where the curves intersect (bounds) |
Steps to Find the Area
- Find the intersection points by solving f(x) = g(x) — these give you a and b
- Determine which function is on top in the interval [a, b]
- Integrate the difference: (top function) - (bottom function)
- If the curves cross within [a, b], split into separate integrals at each crossing point
Horizontal Method
When curves are easier to describe as functions of y, integrate with respect to y instead. Here, f(y) is the right curve and g(y) is the left curve.
Example 1
Find the area between y = x² and y = x
Step 1: Find intersections: x² = x → x² - x = 0 → x(x-1) = 0 → x = 0 and x = 1
Step 2: Between 0 and 1, x > x² (line is above parabola)
Step 3: A = ∫[0 to 1] (x - x²) dx = [x²/2 - x³/3] from 0 to 1
= (1/2 - 1/3) - (0) = 1/6 ≈ 0.1667
Example 2
Find the area between y = x² - 1 and y = 3 - x²
Intersections: x² - 1 = 3 - x² → 2x² = 4 → x = ±√2
Upper curve: 3 - x² is above x² - 1 in this interval
A = ∫[-√2 to √2] [(3-x²) - (x²-1)] dx = ∫[-√2 to √2] (4 - 2x²) dx
= [4x - 2x³/3] from -√2 to √2 = 16√2/3 ≈ 7.54