Integration Concepts and Techniques

These flashcards cover key terms and concepts from integration and its applications in various contexts.

Integral

A mathematical tool used to calculate the area under a curve or the accumulation of quantities.

Antiderivative

A function F(x) whose derivative is f(x); F'(x) = f(x).

Fundamental Theorem of Calculus

Connects differentiation and integration; states that if f is continuous, then ∫[a,b] f(x) dx = F(b) - F(a), where F is any antiderivative of f.

Integration by Substitution

A method to simplify the integration process by substituting a new variable.

Integration by Parts

A technique based on the product rule for differentiation, used for integrating products of functions.

Numerical Integration

The approximation of integral values using numerical techniques rather than analytical expressions.

Double Integral

An integral involving two variables, calculated over a two-dimensional area.

Velocity

The rate of change of position as a function of time.

Displacement

The total distance traveled in a specific direction from the starting point.

Area Under Curve

The integral of a function over an interval, representing the total accumulation of the function's values.

Continuous Function

A function without breaks or jumps, defined for all points in its interval.

Density Function

A function that describes the distribution of mass or probability in a given space.

Trapezoidal Rule

A method for numerical integration that approximates the area under a curve with trapezoids.

Midpoint Rule

A numerical integration method that uses the midpoint of each subinterval for approximating the area.

Error Margin

The difference between the approximate value and the true value in numerical calculations.

Shear Force

The internal force in a beam acting perpendicular to its length.

Bending Moment

The internal moment that induces bending of a beam due to loading.

Modulus of Resilience

The energy per unit volume stored in a material up to the yield point.

Modulus of Toughness

The total energy absorbed by a material until it fractures.

Force Density

Force per unit length or area, often used in load calculations on structures.

Centrifugal Force

The apparent force that draws a rotating body away from the center of rotation.

Parametric Equations

Equations that express a set of quantities as explicit functions of a number of independent parameters.

Piecewise Function

A function defined by different expressions based on the input value.

Maxima and Minima

The largest or smallest values of a function within a given range.

Growth Rate

The rate at which a quantity increases over time.

Velocity-Time Graph

A graph showing the variation of velocity of an object over time.

Constant Velocity

Motion at a fixed speed without acceleration in any direction.

Integration Interval

The range over which an integral is calculated.

Limits of Integration

The specific values that define the start and end of an integration process.

Rectangular Approximations

Estimating area under the curve using rectangles instead of exact calculations.

Diminishing Returns

A decrease in the incremental output or benefit gained from an additional input.

Applied Load

The external forces applied to a structure or material.

Laminate Elements

Composite materials made from layers, often used in construction and design.

Gravitational Force

The force due to gravity acting on an object's mass.

Fluid Mechanics

The study of fluids and how they behave under various forces.

Beam Theory

A theory in structural engineering that describes the behavior of beams under load.

Energy Conservation

A principle stating that energy cannot be created or destroyed, only transformed.

Work-Energy Principle

The concept that the work done on an object is equal to the change in kinetic energy of that object.

Restoring Force

The force that brings a displaced object back to its equilibrium position.

Applied Mechanics

The branch of mechanics that deals with the behavior of solid bodies under external forces.

Mass Centre

The point representing the average position of the mass of a body.

Stress-Strain Curve

A graph that shows the relationship between stress and strain for a material.

Elastic Modulus

A measure of a material's ability to deform elastically (non-permanently) under load.

Deflection

The degree to which a structural element is displaced under load.

Coordinate System

A system that uses numbers to uniquely determine the position of points in a space.

Mathematical Model

A representation of a system using mathematical concepts and language.

Harmonic Motion

The motion of an object that periodically returns to its original position.

Resistance (in circuits)

Opposition to the flow of electric current.

Inductance

A property of an electrically conductive object that opposes changes in electric current.

Capacitance

The ability of a system to store charge per unit voltage.

Thermal Expansion

The tendency of matter to change in volume in response to a change in temperature.

Hydrostatic Pressure

The pressure exerted by a fluid at equilibrium due to the force of gravity.

Material Yield Strength

The stress at which a material begins to deform permanently.

Plastic Deformation

Permanent deformation of a material after the yield point is surpassed.

Convergence (in numerical methods)

The property that a sequence of approximations approaches the exact solution.