# Calculus II For Dummies

Published: 01-24-2012

An easy-to-understand primer on advanced calculus topics

Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams.

It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, with introductions to multivariable calculus, differential equations, and numerical analysis. Best of all, it includes practical exercises designed to simplify and enhance understanding of this complex subject.

• Introduction to integration
• Indefinite integrals
• Intermediate Integration topics
• Infinite series
• Practice exercises

Confounded by curves? Perplexed by polynomials? This plain-English guide to Calculus II will set you straight!

## Articles From Calculus II For Dummies

83 results
83 results
Calculus II For Dummies Cheat Sheet

Cheat Sheet / Updated 08-31-2021

By its nature, calculus can be intimidating. But you can take some of the fear of studying calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Also discover a few basic rules applied to calculus like Cramer's Rule, the Constant Multiple Rule, and a few others, and you'll be on your way to acing the course.

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Calculating Error Bounds for Taylor Polynomials

Article / Updated 07-13-2021

A Taylor polynomial approximates the value of a function, and in many cases, it’s helpful to measure the accuracy of an approximation. This information is provided by the Taylor remainder term: f(x) = Tn(x) + Rn(x) Notice that the addition of the remainder term Rn(x) turns the approximation into an equation. Here’s the formula for the remainder term: It’s important to be clear that this equation is true for one specific value of c on the interval between a and x. It does not work for just any value of c on that interval. Ideally, the remainder term gives you the precise difference between the value of a function and the approximation Tn(x). However, because the value of c is uncertain, in practice the remainder term really provides a worst-case scenario for your approximation. The following example should help to make this idea clear, using the sixth-degree Taylor polynomial for cos x: Suppose that you use this polynomial to approximate cos 1: How accurate is this approximation likely to be? To find out, use the remainder term: cos 1 = T6(x) + R6(x) Adding the associated remainder term changes this approximation into an equation. Here’s the formula for the remainder term: So substituting 1 for x gives you: At this point, you’re apparently stuck, because you don’t know the value of sin c. However, you can plug in c = 0 and c = 1 to give you a range of possible values: Keep in mind that this inequality occurs because of the interval involved, and because that sine increases on that interval. You can get a different bound with a different interval. This simplifies to provide a very close approximation: Thus, the remainder term predicts that the approximate value calculated earlier will be within 0.00017 of the actual value. And, in fact, As you can see, the approximation is within the error bounds predicted by the remainder term.

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Finding the Unit Vector of a Vector

Article / Updated 07-12-2021

Every nonzero vector has a corresponding unit vector, which has the same direction as that vector but a magnitude of 1. To find the unit vector u of the vector you divide that vector by its magnitude as follows: Note that this formula uses scalar multiplication, because the numerator is a vector and the denominator is a scalar. A scalar is just a fancy word for a real number. The name arises because a scalar scales a vector — that is, it changes the scale of a vector. For example, the real number 2 scales the vector v by a factor of 2 so that 2v is twice as long as v. As you may guess from its name, the unit vector is a vector. For example, to find the unit vector u of the vector you first calculate its magnitude |q|: Now use the previous formula to calculate the unit vector: You can check that the magnitude of resulting vector u really is 1 as follows:

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Using the Mean Value Theorem for Integrals

Article / Updated 04-21-2017

The Mean Value Theorem for Integrals guarantees that for every definite integral, a rectangle with the same area and width exists. Moreover, if you superimpose this rectangle on the definite integral, the top of the rectangle intersects the function. This rectangle, by the way, is called the mean-value rectangle for that definite integral. Its existence allows you to calculate the average value of the definite integral. Calculus boasts two Mean Value Theorems — one for derivatives and one for integrals. Here, you will look at the Mean Value Theorem for Integrals. You can find out about the Mean Value Theorem for Derivatives in Calculus For Dummies by Mark Ryan (Wiley). The best way to see how this theorem works is with a visual example: A definite integral and its mean-value rectangle have the same width and area. The first graph in the figure shows the region described by the definite integral This region obviously has a width of 1, and you can evaluate it easily to show that its area is The second graph in the figure shows a rectangle with a width of 1 and an area of It should come as no surprise that this rectangle’s height is also so the top of this rectangle intersects the original function. The fact that the top of the mean-value rectangle intersects the function is mostly a matter of common sense. After all, the height of this rectangle represents the average value that the function attains over a given interval. This value must fall someplace between the function’s maximum and minimum values on that interval. Here’s the formal statement of the Mean Value Theorem for Integrals: If f(x) is a continuous function on the closed interval [a, b], then there exists a number c in that interval such that: This equation may look complicated, but it’s basically a restatement of this familiar equation for the area of a rectangle: Area = Height · Width In other words, start with a definite integral that expresses an area, and then draw a rectangle of equal area with the same width (b – a). The height of that rectangle — f(c) — is such that its top edge intersects the function where x = c. The value f(c) is the average value of f(x) over the interval [a, b]. You can calculate it by rearranging the equation stated in the theorem: For example, here’s a figure that illustrates the definite integral and its mean-value rectangle. Now, here’s how you calculate the average value of the shaded area: Not surprisingly, the average value of this integral is 30, a value between the function’s minimum of 8 and its maximum of 64.

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Using Identities to Express a Trigonometry Function as a Pair of Functions

Article / Updated 04-21-2017

You can express every product of powers of trig functions, no matter how weird, as the product of any pair of trig functions. The three most useful pairings are sine and cosine, tangent and secant, and cotangent and cosecant. The table shows you how to express all six trig functions as each of these pairings. For example, look at the following function: As it stands, you can’t do much to integrate this monster. But try expressing it in terms of each possible pairing of trig functions: As it turns out, the most useful pairing for integration in this case is cot6 x csc2 x. No fraction is present — that is, both terms are raised to positive powers — and the cosecant term is raised to an even power.

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Understanding Power Series

Article / Updated 04-21-2017

The geometric series is a simplified form of a larger set of series called the power series. A power series is any series of the following form: Notice how the power series differs from the geometric series: In a geometric series, every term has the same coefficient. In a power series, the coefficients may be different — usually according to a rule that’s specified in the sigma notation. Here are a few examples of power series: You can think of a power series as a polynomial with an infinite number of terms. For this reason, many useful features of polynomials carry over to power series. The most general form of the power series is as follows: This form is for a power series that’s centered at a. Notice that when a = 0, this form collapses to the simpler version: So a power series in this form is centered at 0.

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Understanding Notations for Sequences

Article / Updated 04-21-2017

Understanding sequences is an important first step toward understanding series. The simplest notation for defining a sequence is a variable with the subscript n surrounded by braces. For example: You can reference a specific term in the sequence by using the subscript: Make sure you understand the difference between notation with and without braces: The notation {an} with braces refers to the entire sequence. The notation an without braces refers to the nth term of the sequence. When defining a sequence, instead of listing the first few terms, you can state a rule based on n. (This is similar to how a function is typically defined.) For example: Sometimes, for increased clarity, the notation includes the first few terms plus a rule for finding the nth term of the sequence. For example: This notation can be made more concise by appending starting and ending values for n: This last example points out the fact that the initial value of n doesn’t have to be 1, which gives you greater flexibility to define a number series by using a rule. Don’t let the fancy notation for number sequences get to you. When you’re faced with a new sequence that’s defined by a rule, jot down the first four or five numbers in that sequence. After you see the pattern, you’ll likely find that a problem is much easier.

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Integrate a Function Using the Tangent Case

Article / Updated 04-21-2017

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Integrate a Function Using the Sine Case

Article / Updated 04-21-2017