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### 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 *n*th-Term Test for Divergence

The *n*th-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 *e**xpressing* a function as an infinite series and *a**pproximating* 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, sin^{3}*x* and cos *x.* This would be simple to differentiate with the Product Rule, but integration doesn’t [more…]