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The Fundamental Theorem of Calculus
Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1
The Indefinite Integral and the Net Change
Indefinite Integrals and Antiderivatives
A Table of Common Antiderivatives
The Net Change Theorem
The NCT and Public Policy
Substitution
Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples
Area Between Curves
Computation Using Integration
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
More Practice
Integration by Parts
Integration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig Functions
Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only
odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases
Trig Substitutions
How Trig Substitution Works
Summary of trig substitution options
Examples
Completing the Square
Partial Fractions
Introduction
Linear Factors
Irreducible 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 1  Improper Integrals with Infinite Intervals of
Integration
Type 2  Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence
Differential Equations
Introduction
Separable Equations
Mixing and Dilution
Models of Growth
Exponential Growth and Decay
Logistic Growth
Infinite Sequences
Approximate Versus Exact Answers
Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence
Infinite Series
Introduction
Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC
Integral Test
Road Map
The Integral Test
Estimates of Value of the Series
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
Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2
Power Series
Radius and Interval of Convergence
Finding the Interval of Convergence
Power Series Centered at $x=a$
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
Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials
Partial Derivatives
Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
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
How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example
Double Integrals over General Regions
Type I and Type II regions
Examples 14
Examples 57
Order of Integration


Volumes as Double Integrals
Double integrals can be viewed as
volumes in the same way that regular
integrals can be viewed as areas.
Previously, we defined the
area under a curve to be an integral, and now we define the volume under a surface
to be a double integral:
Definition:
Let $z=f(x,\,y) \ge 0$ and let $R$ be a rectangle in the
$xy$plane. The double integral $$
\iint_R\, f(x,y)\, dx \,dy $$ of $f$ over $R$ is
the volume of the solid under the graph of $f$ and above
$R$. 
We can compute volumes by computing double integrals. Here , we compute double integrals by using
what we already know about 3dimensional geometry.
Example
1: Compute the integral $\iint_R \frac{4}{3}(5x)
\,dA$, where $R$ is the rectangle $[2,\,5]\,\times\,[1,\,
3]$ in the $xy$plane.
Solution 1: The integral is equal to the volume of
the solid shown here, which we call $W$. $W$ is the
region above $R$ on the $xy$plane and under the (tilted)
plane $z=\frac{4}{3}(5x)$. To find its volume, take a
vertical slice for a fixed $y$ with $1\le y\le 3$. The slice
of the solid on the vertical $xz$plane is the same triangle
for each $y$. We can see from the diagram that the
slice has area $$\frac{1}{2}\times \hbox{base}\times
\hbox{height} \ =\frac12\cdot 3\cdot 4=6$$ Then $W$ has
volume $V$ which is the area of the triangle times the
length of the side. So $V=6\cdot 2 = 12$. We get
$$\iint_R \frac{4}{3}(5x)\, dA = \hbox{ volume of $W$ } =V=
12.$$ 

