# Calculus 2

## Explore Calculus 2 with these articles and videos on integration, infinite series, and other topics that explain (step-by-step) in great detail.

Calculus 2 is a mathematics course for high-school or undergraduate students. The prerequisite for this course is Calculus 1, as it is a direct continuation. This course’s main topics include applications of the definite integral, integration techniques, and a great deal about infinite sequences and series. Other topics that instructors sometimes have are analytic geometry in three dimensions with vector geometry and differential equations. This course is magnificent as students become skilled in integration and learn about improper integrals and Taylor Series.

## Volumes of Solids of Revolution (Disks, Washers, and Shells) [Video]

Finding the volume of the solid generated by rotating a bounded planar region about an axis of rotation is discussed. We cover the disk method, the washer method, and the method of cylindrical shells. We provide several examples of solids generated by revolving around both vertical and horizontal lines.

## Area Between Curves (The Best How to Guide) [Video]

If you already know the Fundamental Theorem of Calculus and find the area under a graph of a function, you are ready to find the area between curves. In this article, I’ll show you how to find the area of a region bounded by two curves. I’ll demonstrate both using vertical strips and horizontal strips.

## Taylor Polynomials and Approximations

By now, you’ve heard of the tangent line approximation. IN this article, I discuss polynomial approximations. I begin by motivating Maclaurin and Taylor and work through several examples.

## Taylor Series (and Maclaurin Series)

In this article, I introduce Taylor Series and Maclaurin Series, and then I discuss the convergence of Taylor Series. After that, I examine the Maclaurin series of the sine and cosine functions and consider precisely when the Maclaurin series for these functions converges.

## Power Series (and Their Convergence)

So now that you understand infinite series, it’s time to dive into power series. In this article, I explain what power series are and discuss the convergence of power series, including the radius of convergence. After that, I cover differentiating and integrating power series and finally combining power series.

## The Ratio Test and the Root Test

The Ratio and Root Tests are criterions for the convergence of an infinite series. I provide several examples using these convergence tests and several exercises.

## Alternating Series Test (and Conditional Convergence)

A series that alternates in sign is called alternating series. Under what conditions will an alternating series converge? Are there any conditions? Also, what does it mean to say or series is conditionally convergent? You will explore absolute and conditional convergence in this article.

## P-Series Test (Theory and Examples)

P-series is one of the most common types of series, and in this article, I go into detail. I explain what the p-series is and work through several examples on how to use the p-series test.

## Direct Comparison Test (and Limit Comparison Test)

One way to determine information about a series is to compare it to another series. This comparison is the idea behind the Direct Comparison Test. Then, I explain the Limit Comparison Test. You will also find many examples.

## Integral Test for Convergence (with Examples)

Yes, it’s possible to determine whether an infinite series is convergent using integration. I convince you that this is natural and that the Integral Test is valuable. You will also see several examples.

## Infinite Series and Convergence

What is an infinite series, and how can you add up an endless amount of numbers and not get infinity? These are the questions I focus on in the article. First, I explain what an infinite series is and then discuss geometric series and the harmonic series. I also cover the divergence test and convergence rules.

## Convergent Sequences (and the Squeeze Theorem)

In this article, I introduce what sequences are, work out several examples, and then explain what a limit of a sequence is. After that, I also discuss the Squeeze Theorem for Limits, monotonic sequences, and bounded sequences.

## Improper Integrals (with Examples)

What’s so improper about an improper integral? Well, there are two ways an integral can be improper. An integral has infinite discontinuities or has infinite limits of integration. I discuss and work through several examples.

## Partial Fractions (Linear and Quadratic Factors)

Partial fractions is an integration technique that allows us to break apart an integrand into fractions. I explain The Method of Partial Fractions, both linear factors, and quadratic factors. You will find many examples.

## Trigonometric Substitution (by Example)

Trigonometric substitution refers to an integration technique that uses trigonometric functions (mostly tangent, sine, and secant) to reduce an integrand to another expression so that one may utilize another known process of integration. I study these three primary forms and give examples to use complete the square to reduce one of these three methods.

## Trigonometric Integrals (Examples Included)

Integrals involving powers of sine and cosine and integrals involving powers of secant and tangent are studied. I also discuss integrals of products of sine and cosine functions involving different angles. Proficiency using trigonometric identities is assumed.

## Integration by Parts (and Reduction Formulas)

Here I motivate and elaborate on an integration technique known as integration by parts. We also demonstrate the repeated application of this formula to evaluate a single integral. The reduction formula for integral powers of the cosine function and an example of its use is also presented.

## Arc Length and Surfaces of Revolution

I motivate an integration formula for finding the arc length of a smooth curve in a plane. Similarly, I use a formula for finding the surface area of the solid obtained by revolving a curve about an axis. After that, I demonstrate these formulas with several examples and provide exercises to develop integration skills.

## Volumes of Solids of Revolution

Finding the volume of the solid generated by rotating a bounded planar region about an axis of rotation is discussed. We cover the disk method, the washer method, and method of cylindrical shells. We provide several examples of solids generated by revolving around both vertical and horizontal lines. Introduction to Solids of Revolution The solid … Read more

## Area Between Two Curves (with Examples)

We explain, through several examples, how to find the area between curves (as a bounded region) using integration. We demonstrate both vertical and horizontal strips and provide several exercises. Introduction to Finding the Area Between Curves When applying the definition for the area between curves, finding the intersection points of the curves and sketching their … Read more