#### A review of integration techniques

We review common techniques to compute indefinite and definite integrals.

#### Area between curves

We introduce the procedure of “Slice, Approximate, Integrate” and use it study the
area of a region between two curves using the definite integral.

#### Accumulated cross-sections

We can also use the procedure of “Slice, Approximate, Integrate” to set up integrals
to compute volumes.

#### What is a solid of revolution?

We define a solid of revolution and discuss how to find the volume of one in two
different ways.

#### The washer method

We use the procedure of “Slice, Approximate, Integrate” to develop the washer
method to compute volumes of solids of revolution.

#### The shell method

We use the procedure of “Slice, Approximate, Integrate” to develop the shell method
to compute volumes of solids of revolution.

#### Length of curves

We can use the procedure of “Slice, Approximate, Integrate” to find the length of
curves.

#### Surface areas of revolution

We compute surface area of a frustrum then use the method of “Slice, Approximate,
Integrate” to find areas of surface areas of revolution.

#### Physical applications

We apply the procedure of “Slice, Approximate, Integrate” to model physical
situations.

#### Integration by parts

We learn a new technique, called integration by parts, to help find antiderivatives of
certain types of products by reexamining the product rule for differentiation.

#### Trigonometric integrals

We can use substitution and trigonometric identities to find antiderivatives of certain
types of trigonometric functions.

#### Improper Integrals

We can use limits to integrate functions on unbounded domains or functions with
unbounded range.

#### Representing sequences visually

We can graph the terms of a sequence and find functions of a real variable that
coincide with sequences on their common domains.

#### The alternating series test

Alternating series are series whose terms alternate in sign between positive and
negative. There is a powerful convergence test for alternating series.

#### Remainders for Geometric and Telescoping Series

For a convergent geometric series or telescoping series, we can find the exact error
made when approximating the infinite series using the sequence of partial
sums.

#### Remainders for alternating series

There is a nice result for approximating the remainder of convergent alternating
series.

#### Remainders and the Integral Test

There is a nice result for approximating the remainder for series that converge by the
integral test.

#### Absolute and Conditional Convergence

The basic question we wish to answer about a series is whether or not the series
converges. If a series has both positive and negative terms, we can refine this question
and ask whether or not the series converges when all terms are replaced by their
absolute values. This is the distinction between absolute and conditional convergence,
which we explore in this section.

#### Higher Order Polynomial Approximations

We can approximate sufficiently differentiable functions by polynomials.

#### Slope fields and Euler’s method

We describe numerical and graphical methods for understanding differential
equations.

#### Separable differential equations

Separable differential equations are those in which the dependent and independent
variables can be separated on opposite sides of the equation.

#### The Dot Product

The dot product is an important operation between vectors that captures geometric
information.