#### How to use Ximera

This course is built in Ximera.

#### How is my work scored?

We explain how your work is scored.

#### A review of differentiation

We review differentiation and integration.

#### A review of integration

We review differentiation and integration.

#### 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.

#### Comparing washer and shell method

We compare and contrast the washer and shell method.

#### 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.

#### Trigonometric substitution

We integrate by substitution with the appropriate trigonometric function.

#### Rational functions

We discuss an approach that allows us to integrate rational functions.

#### Improper Integrals

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

#### Sequences

We investigate sequences.

#### 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.

#### Limits of sequences

There are two ways to establish whether a sequence has a limit.

#### What is a series

A series is an infinite sum of the terms of sequence.

#### Special Series

We discuss convergence results for geometric series and telescoping series.

#### The divergence test

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

#### The integral test

Certain infinite series can be studied using improper integrals.

#### 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.

#### Dig-In: Estimating Series

We learn how to estimate the value of a 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.

#### 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.

#### 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.

#### 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.

#### 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.

#### Parametric equations

We discuss the basics of parametric curves.

#### Calculus and parametric curves

We discuss derivatives of parametrically defined curves.

#### Introduction to polar coordinates

Polar coordinates are coordinates based on an angle and a radius.

#### 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 an important operation between vectors that captures geometric information.

#### Projections and orthogonal decomposition

Projections tell us how much of one vector lies in the direction of another and are important in physical applications.

#### The cross product

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