CalcRef

Applications of Integrals · Area

Area Between Curves

A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\, dx

The area between two curves f(x) and g(x) from x = a to x = b. Take the absolute value or integrate top minus bottom.

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find the area between y = x² and y = x on [0, 1].
  1. On [0,1]: x ≥ x², so top - bottom = x - x²
  2. ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6

Answer: 1/6

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