#### A review of integration

We review differentiation and integration.

#### Area between curves

We compute the area of a region between two curves using the definite integral.

#### Accumulated cross-sections

We can also use integrals to compute volume.

#### Accumulated shells

Some volumes of revolution are more easily computed with cylindrical shells.

#### Length of curves

We can integrate to find the length of curves.

#### Surface area

We compute surface area.

#### Phyical applications

We see several physical applications of integration.

#### Exponential and logarithmetic functions

Exponential and logarithmic functions illuminated.

#### The origins of a logarithm

We look at the origins of a logarithm.

#### Exponential models

We investigate how the exponential functions model phenomena in the real world.

#### Integration by parts

We learn a new technique, called integration by parts, to help us solve problems involving integration.

#### Trigonometric integrals

We can substitution and trigonometric identities to antidifferentiate trigonometric functions.

#### Trigonometric substitution

We integrate by substitution with the appropriate trigonometric function.

#### Rational functions

We can now integrate a large class of functions.

#### Improper Integrals

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

#### Differential equations

Differential equations show you relationships between rates of functions.

#### 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.

#### Sequences

A sequence is an ordered list.

#### Sequences as functions

A function from positive integers to the real numbers is a sequence.

#### Series

A series is summation of a sequence.

#### The integral test

Infinite sums can be studied using improper integrals.

#### The divergence test

If an infinite sum converges, then its terms must tend to zero.

#### The ratio test

Some infinite series can be compared to geometric series.

#### The root test

Some infinite series can be compared to geometric series.

#### The comparison test

We compare infinite series to each other using inequalities.

#### The limit comparison test

We compare infinite series to each other using limits.

#### The alternating series test

Alternating series have nice properties.

#### Approximating functions with polynomials

We can approximate smooth functions with polynomials.

#### Power series

Infinite series can represent functions.

#### Introduction to Taylor series

We study Taylor and Maclaurin series.

#### Numbers and Taylor series

Taylor series are a computational tool.

#### Calculus and Taylor series

Power series interact nicely with other calculus concepts.

#### Parametric equations

We discuss the basics of parametric curves.

#### Calculus and parametric curves

We discuss derivatives and integrals of parametric curves.

#### Introduction to polar coordinates

Polar coordinates are a special type of parametric curves.

#### Gallery of polar curves

We see a collection of polar curves.

#### Derivatives of polar functions

We differentiate polar functions.

#### Integrals of polar functions

We integrate polar functions.

#### Working in two and three dimensions

We talk about basic geometry in higher dimensions.

#### Vectors

Vectors are lists of numbers that denote direction and magnitude.

#### The dot product

The dot product is one way to multiply two vectors.

#### The cross product

The cross product is a special way to multiply two vectors in three-dimensional space.

#### Lines and curves in space

Vector-valued functions are parametrized curves.

#### Calculus and vector-valued functions

With one input, and vector outputs, we work component-wise.