Unit 6 · Integration and Accumulation of Change
● Core concept · ○ Supporting concept
6.1 Exploring Accumulations of Change
Accumulation of change ● (core concept) — The net total change of a quantity over an interval, found by accumulating (integrating) its rate of change.
6.2 Approximating Areas with Riemann Sums
Riemann sum ● (core concept) — An approximation of area under a curve found by summing the areas of rectangles: Σ f(xᵢ*)·Δx over subintervals.
Left Riemann sum ● (core concept) — A Riemann sum using the function value at the left endpoint of each subinterval.
Right Riemann sum ● (core concept) — A Riemann sum using the function value at the right endpoint of each subinterval.
Midpoint Riemann sum ● (core concept) — A Riemann sum using the function value at the midpoint of each subinterval.
Trapezoidal sum ● (core concept) — An area approximation using trapezoids instead of rectangles: each subinterval contributes (Δx/2)·[f(xᵢ₋₁) + f(xᵢ)].
6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
Definite integral ● (core concept) — The exact signed area under f from a to b, defined as the limit of Riemann sums: ∫ₐᵇ f(x) dx = lim(n→∞) Σ f(xᵢ*)·Δx.
Summation (sigma) notation ● (core concept) — The symbol Σ used to write sums compactly: Σ(i=1 to n) aᵢ means a₁ + a₂ + … + aₙ.
6.4 The Fundamental Theorem of Calculus and Accumulation Functions
Fundamental Theorem of Calculus ● (core concept) — Differentiation and integration are inverse processes: if F(x) = ∫ₐˣ f(t) dt for continuous f, then F'(x) = f(x); also ∫ₐᵇ f(x) dx = F(b) − F(a) where F' = f.
6.5 Interpreting the Behavior of Accumulation Functions Involving Area
Accumulation function ● (core concept) — A function defined by A(x) = ∫ₐˣ f(t) dt that gives the accumulated net area under f from a to x.
6.6 Applying Properties of Definite Integrals
Properties of definite integrals ● (core concept) — ∫ₐᵇ [f ± g] = ∫ₐᵇ f ± ∫ₐᵇ g; ∫ₐᵇ c·f = c·∫ₐᵇ f; ∫ₐᵇ f = −∫ᵇₐ f; ∫ₐₐ f = 0; and ∫ₐᵇ f + ∫ᵇᶜ f = ∫ₐᶜ f.
6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Antiderivative ● (core concept) — A function F with F'(x) = f(x); antiderivatives of the same function differ by a constant.
Indefinite integral ● (core concept) — The family of all antiderivatives of f, written ∫ f(x) dx = F(x) + C, where C is the constant of integration.
6.9 Integrating Using Substitution
Integration by substitution ● (core concept) — The reverse Chain Rule: with u = g(x), ∫ f(g(x))·g'(x) dx = ∫ f(u) du.
6.10 Integrating Functions Using Long Division and Completing the Square
Integrating with long division and completing the square ● (core concept) — Rewriting a rational integrand by polynomial long division (or completing the square) into a form whose antiderivative is elementary.
6.11 Integrating Using Integration by Parts
Integration by parts ● (core concept) — A technique for integrating products of functions: the integral of u dv equals u·v minus the integral of v du. Pick u as the part that gets simpler when differentiated and dv as the part you can integrate.
6.12 Integrating Using Linear Partial Fractions
Partial fractions (linear, nonrepeating factors) ● (core concept) — A method for integrating rational functions whose denominator factors into distinct linear factors: split the integrand into a sum of simple fractions like A/(x − a) + B/(x − b) and integrate each term separately.
6.13 Evaluating Improper Integrals
Improper integral ● (core concept) — An integral that has one or both limits infinite, or whose integrand is unbounded somewhere on the interval of integration.
Convergence of an improper integral ● (core concept) — An improper integral is evaluated as a limit of ordinary definite integrals. It converges if that limit is a finite number, and diverges if the limit does not exist or is infinite.