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…]
Knowing When to Integrate by Parts
It’s important to recognize when integrating by parts is useful. To start off, here are two important cases when integration by parts is definitely the way to go: [more…]
How to Integrate Odd Powers of Sines and Cosines
You can integrate any function of the form sinm x cosn xwhen m is odd, for any real value of n. For this procedure, keep in mind the handy trig identity sin [more…]
How to Integrate Even Powers of Sines and Cosines
You can integrate even powers of sines and cosines. For example, if you wanted to integrate sin2 x and cos2 x, you would use these two half-angle trigonometry identities: [more…]
Using Identities to Express a Trigonometry Function as a Pair of Functions
You can express every product of powers of trig functions, no matter how weird, as the product of any pair of trig functions. The three most useful pairings are sine and cosine, tangent and secant, and [more…]
How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules
In many cases, you can untangle hairy rational expressions and integrate them using the anti-differentiation rules plus the Sum Rule, Constant Multiple Rule, and Power Rule. [more…]
When to Use Variable Substitution with Integrals
Variable substitution comes in handy for some integrals. The anti-differentiation formulas plus the Sum Rule, Constant Multiple Rule, and Power Rule allow you to integrate a variety of common functions [more…]
Find the Integral of Nested Functions
Sometimes you need to integrate a function that is the composition of two functions — for example, the function 2x nested inside a sine function. If you were differentiating, you could use the Chain Rule [more…]
Integrate a Function Using the Sine Case
When the function you’re integrating includes a term of the form (a2 – bx2)n, draw your trig substitution triangle for the sine case. For example, suppose that you want to evaluate the following integral [more…]
Integrate a Function Using the Tangent Case
When the function you’re integrating includes a term of the form (a2 + x2)n, draw your trigonometry substitution triangle for the tangent case. For example, suppose that you want to evaluate the following [more…]
Setting Up Partial Fractions When You Have Distinct Factors
Your first step in any problem that involves partial fractions is to recognize which case you’re dealing with so that you can solve the problem. One case where you can use partial fractions is when the [more…]
Setting Up Partial Fractions When You Have Repeated Linear Factors
Your first step in any problem that involves partial fractions is to recognize which case you’re dealing with so that you can solve the problem. One case where you can use partial fractions is with repeated [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…]
