Area Between Curves

Reference for the area between curves formula A = ∫[f(x) - g(x)]dx over [a,b].
Covers vertical vs horizontal slicing and finding intersection points.

The Formula

A = ∫[a to b] |f(x) - g(x)| dx

Where f(x) is the upper curve and g(x) is the lower curve between x = a and x = b.

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)

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

A = ∫[c to d] |f(y) - g(y)| dy

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.

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

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