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 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


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
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
Completing the Square

Partial Fractions

Linear Factors
Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

Differential Equations

Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

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

Infinite Series

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

Alternating Series and the AS Test
Absolute Convergence

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

List of Major Convergence Tests

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

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
Order of Integration
Area and Volume Revisited

One Variable at the Time

Now that we know what double integrals are, we can start to compute them. The key idea is: One variable at a time!

In order to integrate over a rectangle $[a,b] \times [c,d]$, we first integrate over one variable (say, $y$) for each fixed value of $x$. That's an ordinary integral, which we can do using the fundamental theorem of calculus. We then integrate the result over the other variable (in this case $x$), which we can also do using the fundamental theorem of calculus. So a 2-dimensional double integral boils down to two ordinary 1-dimensional integrals, one inside the other. We call this an iterated integral.

There are two ways to see the relation between double integrals and iterated integrals. In the bottom-up approach, we evaluate the sum $$\sum_{i=1}^m \sum_{j=1}^n f\left(x_{i}^*,y_{j}^*\right) \,\Delta x\, \Delta y,$$ by first summing over all of the boxes with a fixed $i$ to get the contribution of a column, and then adding up the columns. (Or we can sum over all of the boxes with a fixed $j$ to get the contribution of a row, and then add up the rows.)

$f\left(x_{i}^*,y_{j}^*\right)\, \Delta y\, \Delta x$ is the approximate contribution of a single box to our double integral.

$\displaystyle\sum_{j=1}^n f\left(x_{i}^*,y_{j}^*\right) \,\Delta y\, \Delta x$ is the approximate contribution of all the boxes in a single column. As $n \to \infty$, the sum over $n$ turns into an integral, and we get $$\displaystyle\left (\int_c^d f\left(x_i^*, y\right)\,dy \right )\, \Delta x.$$

Adding up the columns then gives $\displaystyle\sum_{i=1}^m \left (\int_c^d f\left(x_i^*, y\right)\, dy \right )\, \Delta x$. Taking a limit as $m \to \infty$ turns the sum into an iterated integral: $$\int_a^b \left ( \int_c^d f(x,y)\, dy \right ) \,dx.$$

This bottom-up approach is explained in the following video. (Video Fix? However, there is a small error. At the beginning it says that we're going to integrate over the rectangle $[0,1] \times [0,2]$, but for the rest of the video the region $R$ is actually the rectangle $[0,2] \times [0,1]$.)

Cavalieri's Principle

An alternate approach to finding volumes (and hence double integrals) - the so-called Slice Method - was formulated by Cavalieri and is expressed mathematically in

Cavalieri's Principle: let $W$ be a solid and $P_x,\, a \le x \le b,$ be a family of parallel planes such that
  • $W$ lies between $P_a$ and $P_{\,b}$,
  • the area of the cross-sectional slice of $W$ cut by $P_x$ is $A(x)$.
Then $$\hbox{ volume of} \ W \ = \ \int_a^b\, A(x)\, dx\,.$$

We already used this idea to compute volumes of revolution. Suppose $W$ is created by rotating the graph of $y = f(x),\, a \le x \le b,$ about the $x$-axis. When $P_x$ is a plane perpendicular to the $x$-axis, then the slice of $W$ cut by $P_x$ is a disk of radius $f(x)$. Here $A(x) = \pi f(x)^2$, so we recover the familiar result $$\hbox{ volume of} \ W \ = \ \pi \int_a^b\, f(x)^2\, dx$$ for a volume of revolution. But Cavalieri's Principle does not require the cross-sections to be triangles or disks!

Example: Find the volume of the solid $W$ under the hyperbolic paraboloid $$z \ = \ f(x,\, y) \ = \ 2+ x^2 - y^2 $$ and over the square $D\,= \,[-1,\,1]\,\times\,[-1,\,1]$.

Solution: The solid is shown below.

When $P_x$ is the vertical slice perpendicular to the $x$-axis for fixed $x$ shown in purple, then $$A(x) =\int_{-1}^2\, (2+x^2 - y^2)\, dy\qquad \qquad$$ $$\qquad = \left[\,2y +x^2y -\frac{y^3}{3}\right]_{-1}^1= \frac{10}{3} +2x^2 \,.$$ But then by using the slider to fill out the solid, Cavalieri's Principle shows that $W$ has $$\hbox{volume} \ = \ \int_{-1}^1\, A(x)\, dx \ = \ \int_{-1}^1\, \left(\frac{10}{3} +2x^2\right)\,dx \ = \ \left[\frac{10x}{3} +\frac{2x^3}{3}\right]_{-1}^1 \ = \ 8\,.$$

In other words, the volume of a region is $\int_a^b A(x)\, dx$, where $A(x)$ is the cross-sectional area at a particular value of $x$. But that's the area under the curve $z=f(x,y)$, where we are treating $x$ as a constant and $y$ as our variable. That is,

The double integral $\displaystyle\iint_R f(x,y)\, dA$ equals the iterated integral $\displaystyle\int_a^b \left (\int_c^d f(x,y) \,dy\right )\, dx$.