AP Calculus BC Unit 6 Guidelines: Integration and Accumulation

Accumulation

Adding up infinitesimal changes to find a total.

Net Signed Area

The area between the function f(x) and the x-axis from x=a to x=b.

Definite Integral

Represents the net signed area between a function and the x-axis over a specified interval.

Left Riemann Sum (LRAM)

Approximation that uses the height of the function at the left endpoint of each sub-interval.

Right Riemann Sum (RRAM)

Approximation that uses the height of the function at the right endpoint of each sub-interval.

Midpoint Riemann Sum (MRAM)

Approximation that uses the height of the function at the midpoint of each sub-interval.

Overestimate

An approximation that is greater than the actual value.

Underestimate

An approximation that is less than the actual value.

Trapezoidal Sum

A method of approximation that uses trapezoids rather than rectangles for better accuracy.

Concave Up

A shape where the trapezoidal sum is an overestimate.

Concave Down

A shape where the trapezoidal sum is an underestimate.

Riemann Sum

A method to approximate the area under a curve by summing up areas of rectangles or trapezoids.

Total Change

The difference in the values of an antiderivative at the endpoints of the interval.

Antiderivative

A function F whose derivative is the given function f.

FTC Part 1

States that if f is continuous on [a, b], then the integral of f from a to b equals F(b) - F(a).

FTC Part 2

If g(x) = ∫a^x f(t) dt, then g'(x) = f(x).

U-Substitution

A technique used to simplify integrals by substituting a part of the integral with a new variable.

Integration by Parts

A technique derived from the product rule for differentiation, using the formula ∫u dv = uv - ∫v du.

Linear Property of Integrals

States that ∫(kf(x) ± g(x)) dx = k∫f(x)dx ± ∫g(x)dx.

Improper Integral

An integral with either infinite bounds or discontinuities in the function.

Convergence

When an improper integral has a limit that exists and is finite.

Divergence

When an improper integral's limit is infinite or does not exist.

Primitive

Another term for antiderivative.

Average Value of a Function

Calculated as (1/(b-a))∫a^b f(x) dx.

Average Rate of Change

Calculated as (f(b) - f(a)) / (b - a).

Linearity of Definite Integrals

The property that allows the integral of a sum to be taken as the sum of the integrals.

Additivity of Integrals

The rule that states ∫a^c f(x) dx + ∫c^b f(x) dx = ∫a^b f(x) dx.

Long Division

A technique for simplifying rational functions before integration when the degree of the numerator is higher than that of the denominator.

Completing the Square

A method for rewriting a quadratic expression in a specific form to aid integration.

Table of Values

A set of discrete data used to approximate integrals rather than using a continuous function.

Rectangular Approximation

Estimating the area under a curve by using rectangles.

Partition of Interval

Dividing the interval [a, b] into n smaller sub-intervals for Riemann sums.

Width of Sub-interval

In Riemann sums, denoted as Δx, it is the width of each sub-interval.

Cauchy Criterion

Defines convergence of improper integrals through limit processes.

Limit of Riemann Sum

The approach of Riemann sums as n approaches infinity, used to define definite integrals.

Function Value at Endpoint

The output of a function at a specific point (x=a or x=b) used in Riemann sums.

Negative Area

The area calculated below the x-axis in definite integrals, affecting the total net area.

Secant Line

A line connecting two points on the curve, which relates to the trapezoidal sums.

Geometric Interpretation of Integrals

Understanding the area under a curve visually with geometric shapes.

Fundamental Theorem Connection

The theory that links rates of change (calculus) with total accumulation (integrals).

Vertical Asymptote

A line where a function approaches infinity, affecting the evaluation of integrals.

Differential Calculus

Calculus focused on rates of change and derivatives.

Integral Calculus

Branch of calculus concerned with integrals and accumulation.

Continuity Condition

Requirement for functions to use FTC; the function must be continuous on the closed interval.