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### How to Find Absolute Extrema on a Closed Interval

Every function that’s continuous on a closed interval has an absolute maximum value and an absolute minimum value (the absolute extrema) in that interval — in other words, a highest and lowest point — [more…]

### How to Locate Intervals of Concavity and Inflection Points

You can locate a function's concavity (where a function is concave up or down) and inflection points (where the concavity switches from positive to negative or vice versa) in a few simple steps. The following [more…]

### How to Make Linear Approximations

Because ordinary functions are locally *linear* (that means straight) — and the further you zoom in on them, the straighter they look—a line tangent to a function is a good approximation of the function [more…]

### How Integration Works: It’s Just Fancy Addition

The most fundamental meaning of integration is to add up. And when you depict integration on a graph, you can see the adding up process as a summing up of thin rectangular strips of area to arrive at the [more…]

### How to Approximate Area with Right Rectangles

You can approximate the area under a curve by adding up “right” rectangles. This method works just like the left sum method except that each rectangle is drawn so that its right upper corner touches the [more…]

### How to Find Antiderivatives with Reverse Rules

The easiest antiderivative rules are the ones that are simply the reverse of derivative rules that you probably already know. These rules are automatic, one-step antiderivatives, with the exception of [more…]

### How to Find Antiderivatives by Guessing and Checking

The guess-and-check method works when the *integrand*— that’s the thing you want to antidifferentiate (the expression after the integral symbol, not counting the [more…]

### How to Find Antiderivatives with the Substitution Method

When a function’s argument (that’s the function’s input) is more complicated than something like 3*x* + 2 (a *linear* function of *x* — that is, a function where [more…]

### How to Solve (and Factor) a Quadratic Equation with the Quadratic Formula

A quadratic equation is any *second-degree* polynomial equation — that’s when the highest power of *x,*or whatever other variable is used, is 2. The solution or solutions of a quadratic equation, [more…]

### How to Write Riemann Sums with Sigma Notation

You can use sigma notation to write out the right-rectangle sum for a function. For example, say you’ve got *f*(*x*) = *x*^{2} + 1.

By the way, you don’t need sigma notation for the math that follows. It’s just [more…]

### How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Cosine

When you integrate a trig integral that includes cosine, and if the power of cosine is odd and positive, you can convert and then use substitution to integrate. To make this conversion, you need to know [more…]

### How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity: [more…]

### How to Integrate Tangent/Secant Problems with an Even, Positive Power of Secant

Here’s how you integrate a trig integral that contains tangents and secants where the secant power is even and positive. Like with all tangent/secant integrals, you use the tangent-secant version of the [more…]

### How to Integrate Problems with an Even, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents (and no secant factors) where the tangent power is even and positive. [more…]

### How to Use Trig Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of the following forms (given *a* is a constant and *u* is an expression containing [more…]

### How to Integrate by Using Partial Fractions when the Denominator Contains Only Linear Factors

You can use the partial fractions method to integrate rational functions (Recall that a rational function is one polynomial divided by another.) The basic idea behind the partial fraction approach is “unadding” [more…]

### Integrating Using Partial Fractions when the Denominator Contains Irreducible Quadratic Factors

You can use the partial fractions method to integrate rational functions, including functions with denominators that contain *irreducible* quadratic factors [more…]

### How to Find a Function’s Average Value with the Mean Value Theorem for Integrals

You can find the average value of a function over a closed interval by using the mean value theorem for integrals. The best way to understand the mean value theorem is with a diagram — check it out below [more…]

### How to Find the Volume of a Complicated Shape with the Meat-Slicer Method of Integration

In geometry, you learned how to figure the volumes of simple solids like boxes, cylinders, and spheres. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes [more…]

### 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…]