CALCULUS III
Multiple Integrals
Paul Dawkins
Calculus III
Table of Contents
Preface . ii
Multiple Integrals . 3
Introduction . 3
Double Integrals . 4
Iterated Integrals . 8
Double Integrals Over General Regions .15
Double Int
Area of Plane Regions
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Area of Plane Regions
1 / 13
Outline
1
Finding the Area of a Bounded Region
2
Area Between Two Curves
Math 100 (Inst. of Mathematics)
Area of Plane Regions
2 /
Derivative as Instantaneous Rate of Change
and
Rectilinear Motion
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Derivatives
1 / 27
Outline
1
Derivative as Instantaneous Rate of Change
2
Rectilinear Motion
Math 100 (Inst. of Math
Exponential Growth and Decay and,
Logistic Growth
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Expo and Log Growth
1 / 25
Outline
1
Exponential Growth and Decay
2
Logistic Growth
Math 100 (Inst. of Mathematics)
Expo and Log Gro
Integrals Yielding the Natural Logarithmic Function
and Exponential Function
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
1/1
Recall
Recall the derivative of the natural logarithmic function:
d
1
(ln x) = .
dx
x
Note that ln x
Definite Integrals
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Definite Integrals
1 / 18
Outline
1
Introduction
2
Formal Definition of the Definite Integral
3
Fundamental Theorem of Calculus
4
Properties of Definite Integr
Indeterminate Forms and LHpitals Rule
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Indeterminate Forms and LHpitals Rule
1 / 16
Outline
1
Indeterminate Forms
2
Other Indeterminate Forms
Math 100 (Inst. of Mathematics)
Indetermi
Differential Equations
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Differential Equations
1 / 15
Outline
1
Differential Equations: Definition and Examples
2
Separable Differential Equations
3
Complete Solution to a Differentia
Antiderivatives
and Basic Rules for Integration
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Antiderivatives & Basic Integration Rules
1 / 22
Outline
1
Notations for Antiderivatives
2
Basic Integration Rules
3
The General Power
Integration by Parts
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Integration by Parts
1/1
Outline
Math 100 (Inst. of Mathematics)
Integration by Parts
2/1
Introduction
We have discussed some integration formulas so far and a t
Improper Integrals
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Improper Integrals
1 / 17
Outline
1
Introduction
2
Evaluating Improper Integrals
Type I
Type II
Math 100 (Inst. of Mathematics)
Improper Integrals
2 / 17
Introduct
Functions: Domain and Operations
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Functions and Its Operations
1 / 30
Outline
1
Functions
Domain of Some Functions
Operations on Functions
2
Lecture Exercise
Math 100 (Institute o
Marginal Functions and Related Rates
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Marginal Functions and Related Rates
1 / 28
Outline
1
Marginal Functions
2
Related Rates
Math 100 (Inst. of Mathematics)
Marginal Functions and R
THE DERIVATIVE AND DIFFERENTIATION
A. THE TANGENT LINE AND THE DERIVATIVE
P
For a circle, tangent line
Not necessarily true
intersects at exactly one point.
for all curves.
P
Consider the function f continuous at x1.
secant line any line through two point
Integration by Partial Fractions
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Integration by Partial Fractions
1 / 18
Outline
1
2
Z
p(x)
dx, where deg p < deg q
q(x)
Z
p(x)
dx, where deg p deg q
q(x)
Integrals of the form
Integ
Derivatives of Exponential Functions
and
Derivative as Slope of Tangent Line
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Derivatives
1 / 25
Outline
1
Derivatives of Exponential Function
2
Derivative as Slope of Tangent Line
Ma
Derivatives of Logarithmic Function
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Derivatives of Logarithmic Functions
1 / 27
Outline
1
The Logarithmic Function
2
Derivatives of Logarithmic Functions
3
Examples
4
Logarithmic
Economy in the Long Run:
Classical Theory
Econ 101, 1st Semester, AY 2016-2017
Sarah Lynne S. Daway, Ph.D.
Economy in the Long Run: The
Classical Model
Main Features and Implications:
Long-run model
Supply-determined
Money is neutral (does not affect o
Chapter 4. Life insurance.
Manual for SOA Exam MLC.
Chapter 4. Life insurance.
Actuarial problems.
c
2009.
Miguel A. Arcones. All rights reserved.
Extract from:
Arcones Manual for SOA Exam MLC. Fall 2009 Edition,
available at http:/www.actexmadriver.com/
Some Types of Functions and their Graphs
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Functions
1 / 44
Outline
1
Functions and their Graphs
Graphs of Relations and Functions
Polynomial Functions
Rational Functions
Split Fun
Rules for differentiation
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Rules for differentiation
1 / 16
Recall
The derivative of a function f (x) at x = x0 is given by
f (x0 + h) f (x0 )
.
h
h0
f 0 (x) = lim
Math 100 (Inst. of
Derivatives of Trigonometric Functions
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Derivatives of Trigonometric Functions
1 / 17
Outline
1
Review of Trigonometric Identities
2
Derivatives of Trigonometric Functions
Math 10
Limit of a Function and Limit Theorems
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Limit and Limit Theorems
1 / 23
Outline
1
Limit of a Function
2
Limit Theorems
3
Lecture Exercises
Math 100 (Institute of Mathematics)
Limi
Derivative Tests
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Derivative Tests
1 / 34
Outline
1
Increasing and Decreasing functions
2
Relative and Absolute Extrema
3
Concavity and Points of Inflection
4
Curve Sketching
Math 100
Derivatives of Logarithmic Function
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Derivatives of Logarithmic Functions
1 / 27
Outline
1
The Logarithmic Function
2
Derivatives of Logarithmic Functions
3
Examples
4
Logarithmic
Limits at Infinity
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Limits at Infinity
1 / 15
Outline
1
Limits at Infinity
2
Lecture Exercises
Math 100 (Institute of Mathematics)
Limits at Infinity
2 / 15
Limits at Infinity
2x2
Higher Order Derivatives and Implicit Differentiation
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Higher Order and Implicit Diff
1 / 18
Higher Order Derivatives
Given a differentiable function f (x),
f 0 (x) =
d
[f (x)]
dx
fir
Lines and the Derivative
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Lines and the Derivative
1 / 24
Outline
1
Lines
Equations of a Line
Parallel and Perpendicular Lines
2
The Derivative
Limit Definition of the Derivative
Find
Optimization Problems
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Optimization Problems
1 / 15
Example
Suppose a company has determined that its total revenue R for a
particular product is given by R = x 3 + 450x 2 + 52500x wh
Integrals Yielding the Natural Logarithmic Function
and Exponential Function
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
1 / 16
Recall
Recall the derivative of the natural logarithmic function:
d
1
(ln x) = .
dx
x
Math 100 (In
Area of Plane Regions
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Area of Plane Regions
1 / 13
Outline
1
Finding the Area of a Bounded Region
2
Area Between Two Curves
Math 100 (Inst. of Mathematics)
Area of Plane Regions
2 /
Applications of Differential Equations
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Applications of Differential Equations
1 / 16
Outline
1
Obtaining Equations of Curves given the Slope
2
Rectilinear Motion
3
Marginals
Math 100