Unit 7 AP Calc BC

Net Change Theorem

The total change in a quantity over a given interval, computed as the integral of the rate of change function over that interval.

Velocity Function

A function that describes the rate of change of position with respect to time, often denoted as v(t).

Acceleration Function

The rate of change of velocity, often denoted as a(t), and is the derivative of the velocity function.

Position Function

A function that gives the position of an object at a given time, often denoted as s(t).

Displacement

The net change in position of an object over a given interval, which is the integral of the velocity function.

Rate of Change

The speed at which a quantity changes, expressed as a function of time or another variable.

Work (Integral Definition)

The total work done by a force over a distance, given by the integral of the force function over the distance traveled.

Consumption over Time

The total amount consumed over a period of time, calculated as the integral of the rate of consumption.

Force

A push or pull on an object, often represented as a function that can be integrated to find the work done.

Impulse

The change in momentum of an object, calculated as the integral of force over the time interval.

Work Formula (W = ∫ F(x) dx)

The formula used to calculate work, where F(x) is the force and the integral represents the total work done over a distance.

Area Between Curves

The area enclosed between two curves, calculated by the integral of the difference between the functions over a specified interval.

Definite Integral

An integral that is computed over a specific interval, giving the net area under the curve or the total accumulation of a quantity.

Area Enclosed by Curves

The total area enclosed by two or more intersecting curves, calculated by integrating the difference between the functions.

Intersection of Curves

The points where two curves meet or cross, often found by solving the equation f(x) = g(x).

Integrating with Respect to y

The process of integrating functions where the variable of integration is y instead of x, typically when the curve is expressed in terms of y.

Boundaries with Changing Functions

Problems where the boundaries of the area or volume change dynamically, requiring integration to account for these changes.

Geometric Interpretation of Integrals

Understanding integrals as the area under a curve or the accumulation of a quantity over a certain range.

Function of y (as opposed to x)

A function where the independent variable is y, and integration may be performed with respect to y rather than x.

Volume as an Integral

The volume of a solid can be calculated by integrating the area of cross-sectional slices along a given interval.

Cross Sections

The slices of a solid, taken perpendicular to the axis of integration, whose area is used to calculate the volume.

Square Cross Sections

When the cross sections of a solid are squares, the area of each cross section is A(x) = [f(x)]^2.

Circular Cross Sections

When the cross sections are circles, the area of each cross section is A(x) = π[f(x)]^2.

Cylindrical Shells

A method for finding volume using cylindrical shells, where the formula is V = 2π ∫ a to b x ⋅ f(x) dx.

Volume of Revolution

The volume of a solid generated by revolving a region around an axis, calculated using the disk, washer, or cylindrical shell methods.

Area of Cross Section

The area of a slice of a solid, which is used in volume calculations.

Disk Method

A method for calculating the volume of a solid by integrating the area of circular cross sections.

Washer Method

A method for calculating the volume of a solid by integrating the area of washers (rings) formed by two radii.

Cylindrical Shell Method

A method for calculating volume by using cylindrical shells, which involves integrating around the axis of rotation.

Volume Integral

An integral used to compute the volume of a solid by summing the volumes of infinitesimally small cross-sectional elements.

Arc Length

The total length of a curve between two points, calculated using the formula L = ∫ a to b √[1 + (f'(x))^2] dx.

Curve Length Formula

A formula used to find the length of a curve: L = ∫ a to b √[1 + (f'(x))^2] dx.

Smooth Curve

A curve that is continuous and differentiable, with no sharp corners or cusps.

Sine Wave

A smooth periodic wave represented by the function y = sin(x).

Vertical Tangent

A tangent line that is vertical, occurring when the derivative of the function approaches infinity.

Cusps and Corners

Points on a curve where the derivative is undefined or infinite, resulting in a sharp point.

Derivative of the Curve

The rate of change of the function at any point, often used to calculate arc length.

Work in Physics

The amount of energy transferred by a force, calculated as W = ∫ F(x) dx.

Normal Distribution

A probability distribution that is symmetric about the mean, often used to model natural phenomena.

Probability Density Function (PDF)

A function that describes the likelihood of a random variable taking a particular value, whose integral over an interval gives the probability.

Normal Probabilities

Probabilities calculated from the normal distribution, often using the area under the normal curve.

Expected Value

The mean of a probability distribution, calculated as the integral of x times the probability density function.

Z-Score

A measure of how many standard deviations a value is from the mean, used in standard normal distribution.

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

Work as an Integral

Work can be modeled as the integral of force over distance: W = ∫ F(x) dx.

Definite Integral

An integral computed over a specific interval, providing the accumulated change or area under the curve over that interval.

Indefinite Integral

An integral that represents a family of functions and includes a constant of integration C.

Fundamental Theorem of Calculus

The theorem that relates differentiation and integration, stating that the integral of a function's derivative over an interval gives the net change in the function's values.

Antiderivative

The inverse operation of differentiation, representing the original function from its derivative.

Substitution Method

A method for solving integrals by changing variables to simplify the integral.

Integration by Parts

A method of integration based on the product rule for differentiation, often used for products of functions.

Riemann Sum

A sum used to approximate the value of a definite integral by partitioning the interval and summing the areas of rectangles.

Continuous Function

A function that has no breaks, jumps, or discontinuities over its domain.

Derivative

The rate of change of a function at a particular point, found as the limit of the difference quotient.

Chain Rule for Integration

A rule for differentiating composite functions, which is also used to simplify integrals involving compositions of functions.

Area under a Curve

The total area between a function and the x-axis over a given interval, often found by computing a definite integral.

Upper and Lower Bounds

The limits of integration in a definite integral, representing the start and end points of the interval of interest.

Accumulation Function

A function that gives the total accumulated quantity up to a point, calculated as the integral of a rate of change function.

Midpoint Rule

A numerical integration method that approximates the integral by using the midpoint of each subinterval.

Trapezoidal Rule

A numerical integration method that approximates the area under a curve by dividing the area into trapezoids instead of rectangles.