Indefinite Integral
- Concept: Integration is the reverse process of differentiation.
- Meaning: Represents a family of functions whose derivative gives back the original.
- Graphical Interpretation: Seen as the total accumulation or change of a quantity.
- Applications: Displacement from velocity, calculating potential energy, and modeling population growth.
Integrals of Rational Functions
- Definition: Integration of ratios of polynomial functions.
- Conceptual Understanding: Often broken into simpler parts to make integration possible.
- Applications: Physics (motion equations), economics (growth trends), probability theory.
Integrals of Irrational Functions
- Concept: Integrals involving roots or radicals.
- Understanding Transformations: Simplified into solvable forms through substitutions.
- Applications: Engineering curve design, mechanical models, structural analysis.
Integrals of Trigonometric Functions
- Concept: Integrating sine, cosine, and related functions.
- Visual Understanding: Represented as area under waveforms.
- Applications: Alternating current analysis, oscillations, and sound wave calculations.
Integrals of Hyperbolic Functions
- Concept: Involves hyperbolic sine, cosine, and related functions.
- Applications: Modeling suspension bridges, catenary curves, and heat distribution.
Integrals of Exponential and Logarithmic Functions
- Concept: Involves exponential functions like eˣ and logarithmic functions like ln(x).
- Applications: Population dynamics, radioactive decay, compound interest growth.
Differential Calculus Formulas | Functions, Limits, Derivatives & Applications
Reduction Formulas
- Meaning: Simplifies repeated or complex integrals using patterns.
- Importance: Saves time in higher-order problems.
- Use Cases: Solving integrals of higher powers of trigonometric or polynomial functions.
Definite Integral
- Concept: Represents the net area under a curve between two bounds.
- Interpretation: Measures accumulation within a fixed range.
- Applications: Finding distance, area, volume, or cost over time.
Improper Integral
- Concept: Deals with infinite limits or unbounded functions.
- Understanding Convergence: Determines whether the integral has a finite value.
- Applications: Probability distributions, physical models over infinite ranges.
Double Integral
- Concept: Integration across two variables.
- Interpretation: Measures volume under a surface.
- Applications: Calculating surface area, total mass, and centers of gravity.
Triple Integral
- Concept: Extends integration into three dimensions.
- Interpretation: Determines total volume or mass in 3D regions.
- Applications: Fluid dynamics, physics, and 3D modeling.
Line Integral
- Concept: Integration along a path or curve.
- Meaning: Adds up values along a trajectory.
- Applications: Work done in a force field, path-dependent quantities.
Differential Equations Formulas | First Order, Second Order & Partial Equations
Surface Integral
- Concept: Integration over a surface in space.
- Interpretation: Measures flow or flux through a surface.
- Applications: Electromagnetic theory, fluid dynamics, and field analysis.
