Calculus

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How to Find the Volume of a Circular Shape with the Stack-of-Pancakes Method of Integration

Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. The stack-of-pancakes technique works [more…]

How to Find the Volume of a Shape Using the Washer Method of Integration

Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. If you have a circular shape with [more…]

How to Find the Volume of a Cylindrical Shape with the Nested-Russian-Dolls Method of Integration

Integration enables you to calculate the volumes of an endless variety of complicated shapes that you can’t handle with regular geometry. You can cut up a solid into thin concentric cylinders and then [more…]

How to Calculate Arc Length with Integration

When you use integration to calculate arc length, what you’re doing (sort of) is dividing a length of curve into infinitesimally small sections, figuring the length of each small section, and then adding [more…]

How to Make Unacceptable Forms Acceptable before Using L'Hôpital’s Rule

You can use L'Hôpital’s rule to find a limit when other methods don’t work. In fact, even if some other method does work, L'Hôpital’s rule is often a good shortcut. If substitution of the limit number [more…]

How to Solve Improper Integrals for Functions that Have Vertical Asymptotes

You solve improper integrals by turning them into limit problems. You can’t just do them the regular way. Here’s how you solve improper integrals for functions that have vertical asymptotes. There are [more…]

How to Solve Improper Integrals that Have One or Two Infinite Limits of Integration

One of the ways in which definite integrals can be improper is when one or both of the limits of integration are infinite. You solve this type of improper integral by turning it into a limit problem where [more…]

How to Find the Volume and Surface Area of Gabriel's Horn

Finding the volume and surface area of this horn problem may blow your mind. Gabriel’s horn is the solid generated by revolving about the x-axis the unbounded region between [more…]

How to Do a Related Rate Problem Involving a Moving Baseball

You can use calculus to determine a rate that’s related to the speed of a moving object. For example, say a pitcher delivers a fastball, which the batter pops up — it goes straight up above home plate. [more…]

How to Determine Limits of Sequences with L'Hôpital's Rule

You can use L’Hôpital’s rule to find limits of sequences. L'Hôpital's rule is a great shortcut for when you do limit problems. Here it is: [more…]

How to Use the nth Term Test to Determine Whether a Series Converges

If the individual terms of a series (in other words, the terms of the series’ underlying sequence) do not converge to zero, then the series must diverge. This is the [more…]

How to Work with Geometric Series

Geometric series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A geometric series is a series of the [more…]

How to Analyze a p-Series

So-called p-series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A [more…]

How to Analyze a Telescoping Series

You don’t see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. Consider the following series: [more…]

Using the Direct Comparison Test to Determine If a Series Converges

The direct comparison test is a simple, common-sense rule: If you’ve got a series that’s smaller than a convergent benchmark series, then your series must also converge. And if your series is larger than [more…]

How to Use the Limit Comparison Test to Determine Whether a Series Converges

The idea behind the limit comparison test is that if you take a known convergent series and multiply each of its terms by some number, then that new series also converges. And it doesn’t matter whether [more…]

Determining If a Series Converges Using the Integral Comparison Test

The integral comparison test involves comparing the series you’re investigating to its companion improper integral. If the integral converges, your series converges; and if the integral diverges, so does [more…]

Using the Ratio Test to Determine Whether a Series Converges

The ratio test looks at the ratio of a general term of a series to the immediately preceding term. The ratio test works by looking only at the nature of the series you’re trying to figure out [more…]

How to Use the Root Test to Determine Whether a Series Converges

The root test doesn’t compare a new series to a known benchmark series. It works by looking only at the nature of the series you’re trying to figure out. You use the root test to investigate the limit [more…]

How to Analyze Absolute and Conditional Convergence

Many divergent series of positive terms converge if you change the signs of their terms so they alternate between positive and negative. For example, you know that the harmonic series diverges: [more…]

How to Determine Whether an Alternating Series Converges or Diverges

An alternating series is a series where the terms alternate between positive and negative. You can say that an alternating series converges if two conditions are met: [more…]

How to Find Area with the u-Substitution Method

You can use the Fundamental Theorem to calculate the area under a function (or just to do any old definite integral) that you integrate with the substitution method. What you want to do is to change the [more…]

How to Determine the Dimensions for the Least Expensive Window Frame

You can use calculus to solve practical problems, such as determining the correct size for a home-improvement project. Here’s an example. A Norman window has the shape of a semicircle above a rectangle [more…]

The Difference Quotient: The Bridge between Algebra (Slope) and Calculus (the Derivative)

One of the cornerstones of calculus is the difference quotient. The difference quotient — along with limits — allows you to take the regular old slope formula that you used to compute the slope of lines [more…]

The Definition of the Definite Integral and How it Works

You can approximate the area under a curve by adding up right, left, or midpoint rectangles. To find an exact area, you need to use a definite integral. [more…]

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