Integrals

Integration is the reverse of differentiation and is used to accumulate quantities, such as total distance traveled or area under a curve.

Definite Integrals A definite integral has upper and lower bounds and computes a numerical value:

∫[a to b] f(x) dx = F(b) - F(a)

where F(x) is an antiderivative of f(x).

Area Under a Curve When f(x) ≥ 0 on [a, b], the definite integral gives the area between the curve and the x-axis.

Evaluating Simple Integrals Use the power rule for integration:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1)

For definite integrals, evaluate the antiderivative at the bounds and subtract.

Example: ∫[0 to 2] (2x + 1) dx = [x² + x]₀² = (4 + 2) - (0 + 0) = 6

Relationship to Derivatives The Fundamental Theorem of Calculus links integration and differentiation: if F'(x) = f(x), then ∫[a to b] f(x) dx = F(b) - F(a).

Applications Integrals are used for total accumulated quantities, volumes, probabilities, and area under a curve.