Integration

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.

Double Integral

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

Area Under Curve

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

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.

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.

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.

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.

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.

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.

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.

Convergence (in numerical methods)

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