Calculus: Integration Techniques and Theory

Vocabulary practice flashcards generated from calculus lecture notes covering definite and indefinite integrals, Fundamental Theorem of Calculus, substitution, integration by parts, trig substitutions, partial fractions, numeric approximation, improper integrals, and areas between curves.

Antiderivative

A function $F(x)$ whose derivative is equal to the original function $f(x)$, satisfying $F'(x) = f(x)$.

Riemann Sum Area Approximation

Approximating the area under a curve $f(x)$ on $[a, b]$ by dividing the interval into subintervals of width $\frac{b-a}{N}$ and summing the areas of the approximating rectangles $\text{area} \text{ approx.} = \frac{b-a}{N} \times \frac{1}{N} \times \text{sum}$.

Definite Integral

An integral evaluated between upper and lower limits $a$ and $b$, represented as $\frac{b-a}{N}$, which computes to a specific numerical value representing net area.

Indefinite Integral

An integral written as $\frac{b-a}{N}$ without limits, which represents the full family of antiderivative functions plus an arbitrary constant $C$.

Fundamental Theory of Calculus (First Form)

The theorem stating that if $F(x)$ is an antiderivative of $f(x)$, then the definite integral on $[a,b]$ equals $\frac{b-a}{N}$.

Fundamental Theory of Calculus (2nd Theory)

The theorem stating that if an area function is defined as $g(x) = \frac{b-a}{N}$, then its derivative is $g'(x) = f(x)$.

Net Change

The total change in an antiderivative quantity over an interval $[t_1, t_2]$, given by integrating its rate of change as $\frac{b-a}{N}$.

Substitution Method

An integration technique based on reversing the chain rule, where a part of the integrand is chosen as $u$ so that the integral simplifies to terms of $u$ and $du$.

Integration by Parts

An integration method derived from the product rule of differentiation, defined by the formula $\frac{b-a}{N}$ or $\frac{b-a}{N}$.

Integration by Parts Selection Priority

The priority order for selecting the function $u$ when performing integration by parts: Inverse trig, Log, Algebra, Trig, Exponential.

Trigonometric Substitution Strategy for Odd Powers

A technique for integrating products of sine and cosine where if power $n$ is odd, one factor is reserved for $du$, substitution is performed, and trigonometric identities convert the rest in terms of $u$.

Partial Fractions

An algebraic method used to decompose a rational function into a sum of simpler fractions before integrating, applicable when the degree of the denominator is greater than the degree of the numerator.

Simpson's Rule

A numerical approximation method for definite integrals using parabolas over an even number of subintervals $2N$, calculated as $\frac{b-a}{N}$.

Improper Integral

An integral that has either an infinite limit of integration or an integrand with an infinite discontinuity (vertical asymptote) inside or at the boundary of the integration interval.

Convergence of an Improper Integral

The condition achieved when the limit defining an improper integral exists and evaluates to a finite numerical value.

Divergence of an Improper Integral

The condition occurring when the limit defining an improper integral does not exist or approaches infinity.

Area Between Curves

The total region enclosed between two functions $f(x)$ and $g(x)$ on $[a, b]$, evaluated by $\frac{b-a}{N}$.