Untitled Flashcard Set

Antiderivative

A function F is an antiderivative of f on an interval I when F′(x) = f(x) for all x in I.

Constant of integration (C)

The arbitrary constant added because many functions share the same derivative. It represents the whole family of antiderivatives.

Indefinite integral

The notation ∫ f(x) dx = F(x) + C, denoting the general antiderivative, resulting in a family of functions.

Differential equation

An equation involving a function and its derivatives; finding the antiderivative gives the general solution.

Power Rule

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, where n ≠ −1.

Sigma notation

Σᵢ₌₁ⁿ aᵢ is shorthand for a₁ + a₂ + … + aₙ.

Partition

Dividing [a, b] into n subintervals, each of width Δx = (b − a)/n.

Riemann sum

It's Σ f(cᵢ)Δxᵢ for a partition a = x₀ < x₁ < … < xₙ = b, using sample points cᵢ.

Definite integral

If f is defined on [a, b] and the limit of Riemann sums exists, we write ∫ₐᵇ f(x) dx = lim Σ f(cᵢ)Δxᵢ.

FTC Part 2 (Evaluation Theorem)

If f is continuous on [a, b] and F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

Mean Value Theorem for Integrals

If f is continuous on [a, b], then ∫ₐᵇ f(x) dx = f(c)(b − a) for some c in [a, b].

Average value of f on [a, b]

The number (1/(b − a)) ∫ₐᵇ f(x) dx. This represents the height of a mean value rectangle.

u-substitution

The technique where you set u = g(x), compute du = g′(x) dx, and rewrite the integral in terms of u.

Integration of even functions

On [−a, a], if f is even, ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx.

Integration of odd functions

On [−a, a], if f is odd, ∫₋ₐᵃ f(x) dx = 0.

Additive interval (Thm 4.6)

∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx.

Preservation of inequality (Thm 4.8)

If f(x) ≥ 0 on [a, b], then ∫ₐᵇ f(x) dx ≥ 0.