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Intermediate Integration

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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…]

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…]

How to Evaluate an Improper Integral that Is Vertically Infinite

Improper integrals are useful for solving a variety of problems. A vertically infinite improper integral contains at least one vertical asymptote. Vertically infinite improper integrals are harder to recognize [more…]

How to Find the Volume of a Solid between Two Surfaces of Revolution

If you want to find the volume of a solid that falls between two different surfaces of revolution, you can use the meat-slicer method to do this. The meat-slicer method [more…]

Finding the Area of a Surface of Revolution

The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done.

To find the area of a surface of revolution between [more…]

Using the Shell Method to Find the Volume of a Solid of Revolution

The shell method is useful when you’re measuring a volume of revolution around the y-axis. For example, suppose that you want to measure the volume of the solid shown in this figure. [more…]

Comparing Converging and Diverging Sequences

Every infinite sequence is either convergent or divergent. A convergent sequence has a limit — that is, it approaches a real number. A divergent sequence [more…]

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