Using Trigonometry Substitution to Integrate a Function
Trig substitution allows you to integrate a whole slew of functions that you can’t integrate otherwise. These functions have a special, uniquely scary look about them and are variations on these three [more…]
Integrate a Function Using the Secant Case
When the function that you’re integrating includes a term of the form (bx2 – a2)n, draw your trig substitution triangle for the secant case. For example, suppose that you want to evaluate this integral [more…]
Knowing When to Avoid Trigonometry Substitution
It’s useful to know when you should avoid using trig substitution. With some integrals, it’s better to expand the problem into a polynomial. For example, look at the following integral: [more…]
Setting Up Partial Fractions When You Have Distinct Linear Factors
Your first step in any problem that involves partial fractions is to recognize which case you’re dealing with so that you can solve the problem. The simplest case in which partial fractions are helpful [more…]
How to Distinguish Proper and Improper Rational Expressions
Integration by partial fractions works only with proper rational expressions, but not with improper rational expressions. Telling a proper fraction from an improper one is easy: A fraction a/b is proper [more…]
How to Split One Definite Integral into Two Definite Integrals
When solving area problems, you sometimes need to split an integral into two separate definite integrals. Here’s a simple but handy rule for doing this that looks complicated but is really very easy: [more…]
How to Evaluate an Improper Integral that Is Horizontally Infinite
Improper integrals are useful for solving a variety of problems. A horizontally infinite improper integral contains either ∞ or –∞ (or both) as a limit of integration. [more…]
Find the Area Between Two Functions
To find an area between two functions, you need to set up an equation with a combination of definite integrals of both functions. For example, suppose that you want to calculate the shaded area between [more…]
Using the Mean Value Theorem for Integrals
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, [more…]
Finding the Volume of a Solid with Congruent Cross Sections
Knowing how volume is measured without calculus pays off big-time when you step into the calculus arena. This is strictly no-brainer stuff — some basic, solid geometry that you probably know already. [more…]
Finding the Volume of a Solid with Similar Cross Sections
Finding the volume of a prism or cylinder is pretty straightforward. But what if you need to find the volume of a shape like the one shown here? In this case, slicing parallel to the base always results [more…]
How to Measure the Volume of an Irregular-Shaped Solid
You can measure the volume of any irregular-shaped solid with a cross section that’s a function of x. In some cases, these solids are harder to describe than they are to measure. For example, have a look [more…]
How to Measure the Volume of an Object by Turning It on Its Side
Sometimes if you want to measure the volume of an object, you need to turn it on its side so that you can use the meat-slicer method. This method works best with solids that have similar cross sections [more…]
How to Evaluate the Volume of a Solid of Revolution
You can evaluate the volume of a solid of revolution. A solid of revolution is created by taking a function, or part of a function, and spinning it around an axis — in most cases, either the [more…]
Understanding Notations for Sequences
Understanding sequences is an important first step toward understanding series. The simplest notation for defining a sequence is a variable with the subscript [more…]
Connecting a Series with Its Two Related Sequences
Every series has two related sequences: a defining sequence and a sequence of partial sums. The distinction between a sequence and a series is as follows: [more…]
How to Recognize a P-Series
An important type of series is called the p-series. A p-series can be either divergent or convergent, depending on its value. It takes the following form: [more…]
Using the nth-Term Test for Divergence
The nth-term test for divergence is a very important test, as it enables you to identify lots of series as divergent. Fortunately, it’s also very easy to use. [more…]
How to Integrate a Power Series
Because power series resemble polynomials, they’re simple to integrate using a simple three-step process that uses the Sum Rule, Constant Multiple Rule, and Power Rule. [more…]
Expressing the Function sin x as a Series
If you want to find the approximate value of sin x, you can use a formula to express it as a series. This formula expresses the sine function as an alternating series: [more…]
Expressing the Function cos x as a Series
If you want to find the approximate value of cos x, you start with a formula that expresses the value of sin x for all values of x as an infinite series. Differentiating both sides of this formula leads [more…]
Expressing and Approximating Functions Using the Taylor Series
It’s important to understand the difference between expressing a function as an infinite series and approximating a function by using a finite number of terms of series. You can think of a power series [more…]
Drawing with 3-D Cartesian Coordinates
The three-dimensional (3-D) Cartesian coordinate system (also called 3-D rectangular coordinates) is the natural extension of the 2-D Cartesian graph. The key difference is the addition of a third axis [more…]
Measuring Volume Under a Surface Using a Double Integral
A double integral allows you to measure the volume under a surface as bounded by a rectangle. Definite integrals provide a reliable way to measure the signed area between a function and the [more…]
Evaluating Double Integrals
Double integrals are usually definite integrals, so evaluating them results in a real number. Evaluating double integrals is similar to evaluating nested functions: You work from the inside out. [more…]
