Calculus

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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…]

Evaluating Triple Integrals

Triple integrals are usually definite integrals, so evaluating them results in a real number. Evaluating triple integrals is similar to evaluating nested functions: You work from the inside out. [more…]

Identifying Ordinary, Partial, and Linear Differential Equations

Differential equations (DEs) come in many varieties. And different varieties of DEs can be solved using different methods. You can classify DEs as ordinary and partial Des. In addition to this distinction [more…]

Checking Differential Equation Solutions

Even if you don’t know how to find a solution to a differential equation, you can always check whether a proposed solution works. This is simply a matter of plugging the proposed value of the dependent [more…]

Solving Separable Differential Equations

Differential equations become harder to solve the more entangled they become. In certain cases, however, an equation that looks all tangled up is actually easy to tease apart. Equations of this kind are [more…]

How to Use the Shell Method to Measure the Volume of a Solid

The shell method allows you to measure the volume of a solid by measuring the volume of many concentric surfaces of the volume, called “shells.” Although the shell method works only for solids with circular [more…]

Measuring the Volume of a Pyramid

Suppose that you want to find the volume of a pyramid with a 6-x-6-unit square base and a height of 3 units. Geometry tells you that you can use the following formula: [more…]

Find the Area Under More Than One Function

Sometimes, a single geometric area is described by more than one function. For example, suppose that you want to find the shaded area shown in the following figure, the area under [more…]

Finding the Integral of a Product of Two Functions

Sometimes the function that you’re trying to integrate is the product of two functions — for example, sin3x and cos x. This would be simple to differentiate with the Product Rule, but integration doesn’t [more…]

How to Integrate a Function Multiplied by a Set of Nested Functions

Sometimes you need to integrate the product of a function (x) and a composition of functions (for example, the function 3x2 + 7 nested inside a square root function). If you were differentiating, you could [more…]

How to Integrate Compositions of Functions

Compositions of functions — that is, one function nested inside another — are of the form f(g(x)). You can integrate them by substituting u = g(x) when [more…]

Substituting with Expressions of the Form f(x) Multiplied by g(x)

When g'(x) = f(x), you can use the substitution u = g(x) to integrate expressions of the form f(x) multiplied by g(x). Variable substitution helps to fill the gaps left by the absence of a Product Rule [more…]

Substituting with Expressions of the Form f(x) Multiplied by h(g(x))

When g'(x) = f(x), you can use the substitution u = g(x) to integrate expressions of the form f(x) multiplied by h(g(x)), provided that h is a function that you already know how to integrate. [more…]

Using Variable Substitution to Evaluate Definite Integrals

When using variable substitution to evaluate a definite integral, you can save yourself some trouble at the end of the problem. Specifically, you can leave the solution in terms of [more…]

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