Definite Integration – Limits, Area Under Curve :: Danielitte

Convex Quadrilateral

Segment of Circle

Sector of Circle

Regular Poligon of N Sides

Sperical Segment

Sperical Sector

Frustum of Right Circular Cone

Triangular Prism

Arithmetic Progression

Geometric Progression

Decimal Logarithm

Natural Logarithm

Euler's Formula

Trigonometric Functions For A Right Triangle

Trigonometric Table

Multiple Angle Formulas

Powers Of Trigonometric Functions

Formulas With T=tan(x/2)

Addition Formulas

Sum Of Trigonometric Functions

Product Of Trigonometric Functions

Half Angle Formulas

Angles Of A Plane Triangle

Sides And Angles Of A Plane Triangle

Relationships Among Trigonometric Fuctions

Linear Equation

System of Two Linear Equation

Quadratic Equation

Exponential Equation

Logarithmic Equation

Trigonometric Equation Cos

Trigonometric Equation Sin

Trigonometric Equation Tan

Trigonometric Equation Cotan

Linear Inequation

Quadratic Inequation

Exponential Inequation

Logarithmic Inequation

Trigonometric Inequation Cos

Trigonometric Inequation Sin

Trigonometric Inequation Tan

Trigonometric Inequation Cotan

Horizontal Shifting

Vertical Shifting

Equation of Line

Equation of Circle

Equation of Line Joining Two Points A,B

Equation of Sphere Center at M and Radius R in Rectangular Coordinates

Equation of Ellipsoid With Center M and Semi-Axes A,B,C

Elliptic Cylinder With Axis as Z Axis

Elliptic Cone With Axis as Z Axis

Hyperboloid of One Sheets

Hyperboloid of Two Sheets

Elliptic Paraboloid

Hyperbolic Paraboloid

Differentiation

Indefinite Integrals

Integrals By Partial Functions

Integrals Involving Roots

Integrals Involving Trigonometric Functions

Transformations

Definite Integrals

Transpose Matrix

Addition And Substraction Of Matrices

Multiplication Of Matrices

Determinant Of Matrices

Inverse Of Matrix

Equation In Matrix Form

Properties Of Matrix Calculations

Relative Complement of A in B

Absolute Complement

Symmetric Difference

Operations On Sets

Standard Deviation

Root Mean Square

Normal Distribution(gaussion Distribution)

Exponential Distribution

Poisson Distribution

Uniforn Distribution

Real Form Of Fourier Series

Parseval's Theorem

Fourier Transform

Fourier Symmetry Relationships

Fourier Transform Pairs

Substitution(Frequency Shifting)

Translation(Time Shifting)

Laplace Transform Pairs

Integration - Definite Integrals

Definite Integrals

Fundamental Properties and Applications

A definite integral computes the **net area under a curve** between two specific limits. It is central to many areas of mathematics, science, and engineering.

\[ \int_a^b f(x) \, dx = F(x) \Big|_a^b = F(b) - F(a), \quad F'(x) = f(x). \]
\[ \int_a^b \{ f(x) \pm g(x) \} \, dx = \int_a^b f(x) \, dx \pm \int_a^b g(x) \, dx \]
\[ \int_a^b C f(x) \, dx = C \int_a^b f(x) \, dx \]
\[ \int_a^a f(x) \, dx = 0 \]
\[ \int_a^b f(x) \, dx = - \int_b^a f(x) \, dx \]
\[ \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \]
\[ \int_a^b f(x) \, dx = (b - a) f(c), \quad a \leq c \leq b \]
\[ \int_a^b f(x) g(x) \, dx = f(c) \int_a^b g(x) \, dx, \quad a \leq c \leq b, \quad g(x) \geq 0. \]

Terminology

Definite Integral: Represents the signed area under a curve from \( a \) to \( b \).
Limits of Integration: The lower and upper bounds \( a \) and \( b \).
Net Area: Area above the x-axis is positive; below is negative.
Mean Value Theorem for Integrals: There exists \( c \in [a, b] \) such that \( \int_a^b f(x) dx = (b - a)f(c) \).

Applications

Calculating area between curves and axes.
Determining displacement and distance in physics.
Evaluating probability from continuous probability distributions.
Finding averages and expected values in statistics.
Used in economics for calculating total cost or revenue over time.