The Indefinite Integral And The Net Change

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The Fundamental Theorem of Calculus

Three Different Quantities The Whole as Sum of Partial Changes The Indefinite Integral as Antiderivative The FTC and the Chain Rule

The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives A Table of Common Anti-derivatives The Net Change Theorem The NCT and Public Policy

Substitution

Substitution for Indefinite Integrals Revised Table of Integrals Substitution for Definite Integrals

Area Between Curves

The Slice and Dice Principle To Compute a Bulk Quantity The Area Between Two Curves Horizontal Slicing Summary

Volumes

Slicing and Dicing Solids Solids of Revolution 1: Disks Solids of Revolution 2: Washers Volumes Rotating About the $y$-axis

Integration by Parts

Behind IBP Examples Going in Circles Tricks of the Trade

Integrals of Trig Functions

Basic Trig Functions Product of Sines and Cosines (1) Product of Sines and Cosines (2) Product of Secants and Tangents Other Cases

Trig Substitutions

How it works Examples Completing the Square

Partial Fractions

Introduction Linear Factors Quadratic Factors Improper Rational Functions and Long Division Summary

Strategies of Integration

Substitution Integration by Parts Trig Integrals Trig Substitutions Partial Fractions

Improper Integrals

Type I Integrals Type II Integrals Comparison Tests for Convergence

Differential Equations

Introduction Separable Equations Mixing and Dilution

Models of Growth

Exponential Growth and Decay Logistic Growth

Infinite Sequences

Close is Good Enough (revisited) Examples Limit Laws for Sequences Monotonic Convergence

Infinite Series

Introduction Geometric Series Limit Laws for Series Telescoping Sums and the FTC

Integral Test

Road Map The Integral Test When the Integral Diverges When the Integral Converges

Comparison Tests

The Basic Comparison Test The Limit Comparison Test

Convergence of Series with Negative Terms

Introduction Alternating Series and the AS Test Absolute Convergence Rearrangements

The Ratio and Root Tests

The Ratio Test The Root Test Examples

Strategies for testing Series

List of Major Convergence Tests Examples

Power Series

Radius and Interval of Convergence Finding the Interval of Convergence Other Power Series

Representing Functions as Power Series

Functions as Power Series Derivatives and Integrals of Power Series Applications and Examples

Taylor and Maclaurin Series

The Formula for Taylor Series Taylor Series for Common Functions Adding, Multiplying, and Dividing Power Series Miscellaneous Useful Facts

Applications of Taylor Polynomials

What are Taylor Polynomials? How Accurate are Taylor Polynomials? What can go Wrong? Other Uses of Taylor Polynomials

Partial Derivatives

Definitions and Rules The Geometry of Partial Derivatives Higher Order Derivatives Differentials and Taylor Expansions

Multiple Integrals

Background What is a Double Integral? Volumes as Double Integrals

Iterated Integrals over Rectangles

One Variable at the Time Fubini's Theorem Notation and Order

Double Integrals over General Regions

Type I and Type II regions Examples Order of Integration Area and Volume Revisited

The Net Change Theorem

The net change theorem says that

$$\int_a^b F'(x)\, dx = F(b) - F(a)$$

In other words, the net change in a function is the (definite) integral of its derivative.

In particular, the net distance traveled (final position minus initial position) is the integral of velocity. The net change in velocity (final velocity minus initial velocity) is the integral of acceleration.

Tag » How To Find Net Change