Integration Notation

Direct Answer

Integration notation captures accumulation and serves as the inverse of differentiation. Common forms include ∫ f(x) dx for the antiderivative and ∫_a^b f(x) dx for the signed area under the curve from a to b.

Informal Interpretation

Integration sums infinitely many infinitesimal products f(x) dx, each representing a slice of width dx and height f(x). The total gives area, accumulation, or net change.

Formal Definition

1. Indefinite integral: ∫ f(x) dx = F(x) + C, where F′(x) = f(x) and C is the constant of integration.
2. Definite integral: ∫_a^b f(x) dx = lim_{n → ∞} ∑_i=1ⁿ f(x_i*) Δx, where Δx = (b − a)/n and x_i* ∈ [x_i−1, x_i].

Notational Variants

• Leibniz notation: ∫ f(x) dx, ∫_a^b f(x) dx
• Euler’s inverse operator: D⁻¹[f](x)
• Multiple integrals: ∬ f(x,y) dA, ∭ f(x,y,z) dV
• Line and surface integrals: ∮_C F·dr, ∬_S F·dS

Examples

1. ∫ x² dx = x³/3 + C
2. ∫₀¹ x² dx = [x³/3]₀¹ = 1/3
3. ∫ 2x e^x2 dx = e^x2 + C
4. ∫ u dv = u v − ∫ v du

Common Conventions

• Add + C for any indefinite integral.
• Use [F(x)]_a^b to denote F(b) − F(a).
• Include dx to specify the integration variable.
• Omit the differential when context makes the variable clear.

Next Steps

1. Study the Fundamental Theorem of Calculus linking differentiation and integration.
2. Learn techniques: substitution, integration by parts, partial fractions.
3. Explore applications: areas, volumes, arc length, and mean value.
4. Dive into vector calculus: Green’s, Stokes’, and divergence theorems.