Comprehensive Guide to Integration and Accumulation

Derivative

The instantaneous rate of change of a function.

Integral

The accumulation of change over an interval; it is the antiderivative denoted by $B$ \int \$.

Definite Integral

Represents the net signed area bounded by the graph of a function and the x-axis between two x-values.

Positive Area

The area above the x-axis.

Negative Area

The area below the x-axis.

Riemann Sum

A method to estimate the area under a curve using shapes like rectangles.

Left Riemann Sum (LRAM)

Uses the left endpoint of subintervals to determine rectangle heights.

Right Riemann Sum (RRAM)

Uses the right endpoint of subintervals to determine rectangle heights.

Midpoint Riemann Sum (MRAM)

Uses the midpoint of subintervals to determine rectangle heights.

Trapezoidal Sum

An approximation method that uses trapezoids instead of rectangles to estimate area under a curve.

Area of a Trapezoid Formula

A = \frac{1}{2}(b1 + b2)h, where b1 and b2 are bases and h is the height.

Indefinite Integral

Integral without limits representing a family of functions whose derivative is given.

Power Rule for Integration

\int x^n dx = \frac{x^{n+1}}{n+1} + C where n \neq -1.

Constant of Integration (C)

A constant added to an indefinite integral because the exact constant value is unknown.

Fundamental Theorem of Calculus

Connects differential calculus and integral calculus.

Evaluation Theorem

\int_a^b f(x) dx = F(b) - F(a) where F is an antiderivative of f.

Zero Width Property

\int_a^a f(x) dx = 0.

Reversal Property

\inta^b f(x) dx = -\intb^a f(x) dx.

Additivity Property

\inta^c f(x) dx = \inta^b f(x) dx + \int_b^c f(x) dx.

Linearity Property

\inta^b [k \cdot f(x) + g(x)] dx = k\inta^b f(x)dx + \int_a^b g(x)dx.

Method of Substitution (U-Sub)

A technique for integration that involves substituting a composite function with a new variable.

Chain Rule in FTC

For\ \frac{d}{dx}\int_a^{g(x)} f(t) dt = f(g(x)) \cdot g'(x).

Trapezoidal Rule

An approximation technique that uses the average of heights of function values at both ends of an interval.

Limit of Riemann Sums

The limit of the sum of the areas of rectangles as the number of rectangles approaches infinity.

Accumulation

The total change over an interval, related to definite integrals.

Net Signed Area

The total area calculated considering positive and negative contributions based on the position relative to the x-axis.

Continuous Function

A function that is unbroken and has no gaps on the interval of integration.

Integrable Functions

Functions that can be integrated over a specified interval.

Exam Pitfall: Forgetting C

A common mistake where the constant of integration is omitted in indefinite integrals.

Exam Pitfall: Uneven Tables

Students may assume that the widths of intervals are constant when calculating Riemann sums.

Exam Pitfall: Trapezoid Area Calculation

Making errors in calculating the area of trapezoids when the widths vary.

Displacement vs. Distance

\inta^b v(t) dt gives net change (displacement); \inta^b |v(t)| dt gives total distance traveled.

Antiderivative

A function whose derivative is the given function, crucial in calculating integrals.

Geometric Interpretation of Integrals

The integral's value corresponds to the area under the curve of a function.

Integration Techniques

Methods such as substitution, integration by parts, and recognizing patterns to simplify integrals.

Holding Area Constant

An important aspect of trapezoidal sums and other approximations, where widths may not vary.

Function Composition

The operation of applying one function to the result of another function, often seen in u-substitution.

Finite Differences

Values calculated between different intervals or data points; crucial for approximating integrals from data tables.

Cumulative Distribution Function

An example from probability theory that utilizes integrals to determine probabilities.

Approximate Integration Techniques

Methods to estimate the value of definite integrals, especially when exact values are difficult to obtain.

Bounds Change in U-Sub

When performing definite integrals with u-substitution, the limits of integration must also change accordingly.

Signs in Integration

Considering the signs of areas in definite integrals, highlighting when to subtract negative areas.

Height of Rectangles in Riemann Sums

Determined by the y-value of the function at specified points (left, right, or midpoint).

Limits in Calculus

Foundational concept in calculus determining the behavior of functions as inputs approach a specific value.