Calculus II For Dummies
Overview
The easy (okay, easier) way to master advanced calculus topics and theories
Calculus II For Dummies will help you get through your (notoriously difficult) calc class—or pass a standardized test like the MCAT with flying colors. Calculus is required for many majors, but not everyone’s a natural at it. This friendly book breaks down tricky concepts in plain English, in a way that you can understand. Practical examples and detailed walkthroughs help you manage differentiation, integration, and everything in between. You’ll refresh your knowledge of algebra, pre-calc and Calculus I topics, then move on to the more advanced stuff, with plenty of problem-solving tips along the way.
- Review Algebra, Pre-Calculus, and Calculus I concepts
- Make sense of complicated processes and equations
- Get clear explanations of how to use trigonometry functions
- Walk through practice examples to master Calc II
Use this essential resource as a supplement to your textbook or as refresher before taking a test—it’s packed with all the helpful knowledge you need to succeed in Calculus II.
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About The Author
Mark Zegarelli is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II Workbook For Dummies. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com.
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calculus ii for dummies
CHEAT SHEET
By its nature, calculus can be intimidating. But you can take some of the fear of studying calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Also discover a few basic rules applied to calculus like Cramer's Rule, the Constant Multiple Rule, and a few others, and you'll be on your way to acing the course.
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You can add and subtract vectors on a graph by beginning one vector at the endpoint of another vector. You add and subtract vectors component by component, as follows:
When you add or subtract two vectors, the result is a vector.
Geometrically speaking, the net effects of vector addition and subtraction are shown here.
Calculus
A Taylor polynomial approximates the value of a function, and in many cases, it’s helpful to measure the accuracy of an approximation. This information is provided by the Taylor remainder term:f(x) = Tn(x) + Rn(x)Notice that the addition of the remainder term Rn(x) turns the approximation into an equation. Here’s the formula for the remainder term:It’s important to be clear that this equation is true for one specific value of c on the interval between a and x.
Vectors are commonly used to model forces such as wind, sea current, gravity, and electromagnetism. Calculating the magnitude of vectors is essential for all sorts of problems where forces collide.
Magnitude is defined as the length of a vector. The notation for absolute value
is also used for the magnitude of a vector.
By its nature, calculus can be intimidating. But you can take some of the fear of studying calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Also discover a few basic rules applied to calculus like Cramer's Rule, the Constant Multiple Rule, and a few others, and you'll be on your way to acing the course.
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 variable into both sides of the equation to see whether equality is maintained.
For example, here’s a differential equation with a dependent variable y:
You may not have a clue how to begin solving this differential equation, but imagine that an angel lands on your pen and offers you this solution:
y = 4e3x sin x
You can check to see whether this angel really knows math by plugging in this value of y as follows:
Because the left and right sides of the equation are equal, the angel’s solution checks out.
Every infinite sequence is either convergent or divergent. A convergent sequence has a limit — that is, it approaches a real number. A divergent sequence doesn’t have a limit.
Here’s an example of a convergent sequence:
This sequence approaches 0, so:
Thus, this sequence converges to 0.
Here’s another convergent sequence:
This time, the sequence approaches 8 from above and below, so:
In many cases, however, a sequence diverges — that is, it fails to approach any real number.
Calculus
In trying to understand what makes a function integrable, you first need to understand two related issues: difficulties in computing integrals and representing integrals as functions.
Computing integrals
For many input functions, integrals are more difficult to compute than derivatives. For example, suppose that you want to differentiate and integrate the following function:
y = 3x5e2x
You can differentiate this function easily by using the Product Rule:
Because no such rule exists for integration, in this example you’re forced to seek another method.
Calculus
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:
A sequence is a list of numbers separated by commas (for example: 1, 2, 3, ...).
A series is a sum of numbers separated by plus signs (for example: 1 + 2 + 3 + .
The solution to a definite integral gives you the signed area of a region. In some cases, signed area is what you want, but in some problems you’re looking for unsigned area.
The signed area above the x-axis is positive, but the signed area below the x-axis is negative. In contrast, unsigned area is always positive.
You can use the concept of unsigned area to measure the area between curves. For example, you can use this technique to find the unsigned shaded area in the following figure.
Finding the area between y = 4x – x2 and y = sin x from x = 0 to x = 4.
In this example, you will approximate your answer to two decimal places using
The first step is to find an equation for the solution (which will probably give you partial credit), and then worry about solving it.
Calculus
Because the Taylor series is a form of power series, every Taylor series also has an interval of convergence. When this interval is the entire set of real numbers, you can use the series to find the value of f(x) for every real value of x.
However, when the interval of convergence for a Taylor series is bounded — that is, when it diverges for some values of x — you can use it to find the value of f(x) only on its interval of convergence.
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, the z-axis, extending perpendicularly through the origin.
The first octant of the 3-D Cartesian coordinate system.
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.
You can solve double integrals in two steps: First evaluate the inner integral, and then plug this solution into the outer integral and solve that.
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.
Triple integrals look scary, but if you take them step by step, they’re no more difficult than regular integrals. You start in the center and work your way out.
Calculus
The Maclaurin series is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin x and cos x as infinite series.
Without further ado, here it is:
The notation f(n) means “the nth derivative of f.” This becomes clearer in the expanded version of the Maclaurin series:
The Maclaurin series allows you to express functions as power series by following these steps:
Find the first few derivatives of the function until you recognize a pattern.
Calculus
The Taylor series provides a template for representing a wide variety of functions as power series. It is relatively simple to work with, and you can tailor it to obtain a good approximation of many functions.
Here’s the Taylor series in all its glory:
The Taylor series uses the notation f(n) to indicate the nth derivative.
Calculus
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 as a polynomial with infinitely many terms (Taylor polynomial).
Every Taylor series provides the exact value of a function for all values of x where that series converges.
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 to a similar formula for cos x:
Now evaluate these derivatives:
Finally, simplify the result a bit:
As you can see, the result is a power 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:
To make sense of this formula, use expanded notation:
Notice that this is a power series. To get a quick sense of how it works, here’s how you can find the value of sin 0 by substituting 0 for x:
As you can see, the formula verifies what you already know: sin 0 = 0.
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 y = x2 and
as shown in this figure.
First, notice that the two functions y = x2 and
intersect where x = 1. This information is important because it enables you to set up two definite integrals to help you find region A:
Although neither equation gives you the exact information that you’re looking for, together they help you out.
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 y = sin x and y = cos x from 0 to Π/2:
The first thing to notice is that the shaded area isn’t under a single function, so you can’t expect to use a single integral to find it.
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. Unfortunately, no Chain Rule exists for integration.
Fortunately, a function such as
is a good candidate for variable substitution.
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 a and b, watch this video tutorial or follow the steps below:
This formula looks long and complicated, but it makes more sense when you spend a minute thinking about it.
Calculus
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 have a Product Rule. Fortunately, variable substitution comes to the rescue.
Given the example,
follow these steps:
Declare a variable as follows and substitute it into the integral:
Let u = sin x
You can substitute this variable into the expression that you want to integrate as follows:
Notice that the expression cos x dx still remains and needs to be expressed in terms of u.
Every nonzero vector has a corresponding unit vector, which has the same direction as that vector but a magnitude of 1. To find the unit vector u of the vectoryou divide that vector by its magnitude as follows:Note that this formula uses scalar multiplication, because the numerator is a vector and the denominator is a scalar.
Calculus
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.
One of the simplest solids to find the volume of is a prism. A prism is a solid that has all congruent cross sections in the shape of a polygon.
Calculus
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 in the same shape — a circle — but the area may differ. That is, the solid has similar cross sections rather than congruent ones.
Calculus
The anti-differentiation rules for integrating greatly limit how many integrals you can compute easily. In many cases, however, you can integrate any polynomial in three steps by using the Sum Rule, Constant Multiple Rule, and Power Rule.
Here’s how:
Use the Sum Rule to break the polynomial into its terms and integrate each of these separately.
Calculus
Integration by partial fractions works only with proper rational expressions, but not with improper rational expressions. Telling a proper fraction from an improper one is easy: A fraction a/b is proper if the numerator (disregarding sign) is less than the denominator, and improper otherwise.
With rational expressions, the idea is similar, but instead of comparing the value of the numerator and denominator, you compare their degrees.
Calculus
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.
Evaluating an improper integral is a three-step process:
Express the improper integral as the limit of a proper integral.
Evaluate the integral by whatever method works.
Calculus
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 than those that are horizontally infinite. An integral of this type contains at least one vertical asymptote in the area that you’re measuring.
Calculus
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 x-axis or the y-axis.
A solid of revolution of y = 2 sin x around the x-axis.
For example, the left side of the figure shows the function y = 2 sin x between x = 0 and
Every solid of revolution has circular cross sections perpendicular to the axis of revolution.
Calculus
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 works best with solids that have similar cross sections. Here’s the plan:
Find an expression that represents the area of a random cross section of the solid in terms of x.
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
You know how to integrate the outer function f.
The inner function g(x) differentiates to a constant — that is, it’s of the form ax or ax + b.
Here’s an example.
Calculus
You can integrate even powers of secants with tangents. If you wanted to integrate tanm x secn x when n is even — for example, tan8x sec6x — you would follow these steps:
Peel off a sec2x and place it next to the dx:
Use the trig identity 1 + tan2 x = sec2x to express the remaining secant factors in terms of tangents:
Use the variable substitution u = tan x and du = sec2x dx:
At this point, the integral is a polynomial, and you can evaluate it.
Calculus
You can integrate even powers of sines and cosines. For example, if you wanted to integrate sin2x and cos2x, you would use these two half-angle trigonometry identities:
Here’s how you integrate cos2x:
Use the half-angle identity for cosine to rewrite the integral in terms of cos 2x:
Use the Constant Multiple Rule to move the denominator outside the integral:
Distribute the function and use the Sum Rule to split it into several integrals:
Evaluate the two integrals separately:
As a second example, here’s how you integrate sin2 x cos4 x:
Use the two half-angle identities to rewrite the integral in terms of cos 2x:
Use the Constant Multiple Rule to move the denominators outside the integral:
Distribute the function and use the Sum Rule to split it into several integrals:
Evaluate the resulting odd-powered integrals:
With the integration behind you, use algebra to simplify the result.
Calculus
You can integrate any function of the form sinm x cosn x when m is odd, for any real value of n. For this procedure, keep in mind the handy trig identity sin2x + cos2x = 1. For example, here’s how you integrate
Peel off a sin x and place it next to the dx:
Apply the trig identity sin2x = 1 – cos2x to express the rest of the sines in the function as cosines:
Use the variable substitution u = cos x and du = –sin x dx:
Now that you have the function in terms of powers of u, the worst is over.
Calculus
You can integrate odd powers of tangents with secants. To integrate tanm x secn x when m is odd — for example, tan7x sec9x — you would follow these steps:
Peel off a tan x and a sec x and place them next to the dx:
Use the trig identity tan2x = sec2x – 1 to express the remaining tangent factors in terms of secants:
Use the variable substitution u = sec x and du = sec x tan x dx:
At this point, the integral is a polynomial, and you can evaluate it.
Calculus
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.
The Sum Rule for integration tells you that integrating long expressions term by term is okay. Here it is formally:
The Constant Multiple Rule tells you that you can move a constant outside of a derivative before you integrate.
Calculus
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 use a combination of the Product Rule and the Chain Rule, but these options aren’t available for integration.
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.
For example, take a look at the following integral:
At first glance, this integral of a series may look scary. But to give it a chance to sho
Calculus
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 at this figure.
A solid based on two exponential curves in space.
The solid in the figure consists of two exponential curves — one described by the equation y = ex and the other described by placing the same curve directly in front of the x-axis — joined by straight lines.
Calculus
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.
Here’s the plan:
Find an expression that represents the area of a random cross section of the solid in terms of x.
Cylindrical coordinates are simply polar coordinates with the addition of a vertical z-axis extending from the origin. While a polar coordinate pair is of the form
with cylindrical coordinates, every point in space is assigned a set of coordinates of the form
The polar coordinate system assigns a pairing of values to every point on the plane.
Spherical coordinates are used — with slight variation — to measure latitude, longitude, and altitude on the most important sphere of them all, the planet Earth. Every point in space is assigned a set of spherical coordinates of the form
In case you’re not in a sorority or fraternity,
is the lowercase Greek letter rho,
is the lowercase Greek letter theta (commonly used in math to represent an angle),
is the lowercase Greek letter phi,
which is commonly pronounced either “fee” or “fye” (but never “foe” or “fum”).
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:
Here’s a common example of a p-series, when p = 2:
Here are a few other examples of p-series:
Remember not to confuse p-series with geometric series. Here’s the difference:
A geometric series has the variable n in the exponent — for example,
A p-series has the variable in the base — for example
As with geometric series, a simple rule exists for determining whether a p-series is convergent or divergent.
Calculus
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:
This rule just says that you can split an area into two pieces and then add up the pieces to get the area that you started with.
Calculus
You’ll be surprised how much headway you can often make when you integrate an unfamiliar trigonometry function by first tweaking it using the Basic Five trig identities:
The unseen power of these identities lies in the fact that they allow you to express any combination of trig functions into a combination of sines and cosines.
Calculus
You can use a partial derivative to measure a rate of change in a coordinate direction in three dimensions. To do this, you visualize a function of two variables z = f(x, y) as a surface floating over the xy-plane of a 3-D Cartesian graph. The following figure contains a sample function.
Now take a look at the function z = y, shown here.
Calculus
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 cross sections, it’s ideal for solids of revolution around the y-axis, because you don’t have to use inverses of functions.
Calculus
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 they can be further distinguished by their order.Here are some examples:Solving a differential equation means finding the value of the dependent variable in terms of the independent variable.
When the function that you’re integrating includes a term of the form (bx2 – a2)n, draw your trig substitution triangle for the secant case. For example, suppose that you want to evaluate this integral:
This is a secant case, because a multiple of x2 minus a constant is being raised to a power
Integrate using trig substitution as follows:
Draw the trig substitution triangle for the secant 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:This is a sine case, because a constant minus a multiple of x2 is being raised to a powerHere’s how you use trig substitution to handle the job:
Draw the trig substitution triangle for the correct 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 integral:
This is a tangent case, because a constant plus a multiple of x2 is being raised to a power (–2).
Calculus
You can integrate powers of cotangents and cosecants similar to the way you do tangents and secant. For example, here’s how to integrate cot8x csc6x:
Peel off a csc2x and place it next to the dx:
Use the trig identity 1 + cot2 x = csc2x to express the remaining cosecant factors in terms of cotangents:
Use the variable substitution u = cot x and du = –csc2x dx:
At this point, the integral is a polynomial, and you can evaluate it.
Calculus
It’s useful to know when you should avoid using trig substitution. With some integrals, it’s better to expand the problem into a polynomial. For example, look at the following integral:
This may look like a good place to use trig substitution, but it’s an even better place to use a little algebra to expand the problem into a polynomial:
Similarly, look at this integral:
You can use trig substitution to evaluate this integral if you want to.
Calculus
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 x-axis as bounded by any two values of x. Similarly, a double integral allows you to measure the signed volume between a function z = f(x, y) and the xy-plane as bounded by any two values of x and any two values of y.
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:This formula works just fine, but it doesn’t give you insight into how to solve similar problems; it works only for pyramids. The meat-slicer method, however, provides an approach to the problem that you can generalize to use for many other types of solids.
Calculus
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 denominator is the product of distinct quadratic factors — that is, quadratic factors that are nonrepeating.
For each distinct quadratic factor in the denominator, add a partial fraction of the following form:For example, suppose that you want to integrate this function:The first factor in the denominator is linear, but the second is quadratic and can’t be decomposed to linear factors.
Calculus
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. The simplest case in which partial fractions are helpful is when the denominator is the product of distinct linear factors — that is, linear factors that are nonrepeating.
Calculus
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 linear factors. These are difficult to work with because each factor requires more than one partial fraction.
Calculus
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 quadratic factors.This is your worst nightmare when it comes to partial fractions, because the denominator includes repeated quadratic factors.
Calculus
A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. This method involves multiplying the entire equation by an integrating factor. A linear first-order equation takes the following form:
To use this method, follow these steps:
Calculate the integrating factor.
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 called separable equations (or autonomous equations), and they fit into the following form:
Separable equations are relatively easy to solve.
Calculus
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 and a Chain Rule for integration.
Some products of functions yield quite well to variable substitution. Look for expressions of the form f(x) multiplied by g(x) where
You know how to integrate g(x).
Calculus
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.Variable substitution helps to fill the gaps left by the absence of a Product Rule and a Chain Rule for integration.Here’s a hairy-looking integral that actually responds well to substitution:The key insight here is that the numerator of this fraction is the derivative of the inner function in the denominator.
Understanding sequences is an important first step toward understanding series. The simplest notation for defining a sequence is a variable with the subscript n surrounded by braces. For example:
You can reference a specific term in the sequence by using the subscript:
Make sure you understand the difference between notation with and without braces:
The notation {an} with braces refers to the entire sequence.
The geometric series is a simplified form of a larger set of series called the power series. A power series is any series of the following form:
Notice how the power series differs from the geometric series:
In a geometric series, every term has the same coefficient.
In a power series, the coefficients may be different — usually according to a rule that’s specified in the sigma notation.
Calculus
When mathematicians discuss whether a function is integrable, they aren’t talking about the difficulty of computing that integral — or even whether a method has been discovered. Each year, mathematicians find new ways to integrate classes of functions. However, this fact doesn’t mean that previously nonintegrable functions are now integrable.
Unlike geometric series and p-series, a power series often converges or diverges based on its x value. This leads to a new concept when dealing with power series: the interval of convergence.
The interval of convergence for a power series is the set of x values for which that series converges.
The interval of convergence is never empty
Every power series converges for some value of x.
Calculus
You can use a shortcut to integrate compositions of functions — that is, nested functions of the form f(g(x)). Technically, you’re using the variable substitution u = g(x), but you can bypass this step and still get the right answer.
This shortcut works for compositions of functions f(g(x)) for which
You know how to integrate the outer function f.
Calculus
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 cotangent and cosecant. The table shows you how to express all six trig functions as each of these pairings.
For example, look at the following function:
As it stands, you can’t do much to integrate this monster.
Multiplying a vector by a scalar is called scalar multiplication. To perform scalar multiplication, you need to multiply the scalar by each component of the vector.
A scalar is just a fancy word for a real number. The name arises because a scalar scales a vector — that is, it changes the scale of a vector. For example, the real number 2 scales the vector v by a factor of 2 so that 2v is twice as long as v.
Calculus
Trig substitution allows you to integrate a whole slew of functions that you can’t integrate otherwise. These functions have a special, uniquely scary look about them and are variations on these three themes:
(a2 – bx2)n
(a2 + bx2)n
(bx2 – a2)n
Trig substitution is most useful when n is
or a negative number — that is, for hairy square roots and polynomials in the denominator of a fraction.
Calculus
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 u by changing the limits of integration.
For example, suppose that you’re evaluating the following definite integral:
Notice that this example gives the limits of integration as x = 0 and x = 1.
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.
If the limit of sequence {an} doesn’t equal 0, then the series ∑ an is divergent.
To show you why this test works, the following sequence meets the necessary condition — that is, a sequence that doesn’t approach 0:
Notice that the limit of the sequence is 1 rather than 0.
Calculus
The Constant Multiple Rule for integration allows you to simplify an integral by factoring out a constant. This option is also available when you’re working with series. Here’s the rule:
For example:
To see why this rule works, first expand the series so you can see what you’re working with:
Working with the expanded form, you can factor a 7 from each term:
Now express the contents of the parentheses in sigma notation:
As if by magic, this procedure demonstrates that the two sigma expressions are equal.
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, the top of the rectangle intersects the function. This rectangle, by the way, is called the mean-value rectangle for that definite integral.
Calculus
The Product Rule enables you to integrate the product of two functions. For example, through a series of mathematical somersaults, you can turn the following equation into a formula that’s useful for integrating.
This derivation doesn’t have any truly difficult steps, but the notation along the way is mind-deadening, so don’t worry if you have trouble following it.
Calculus
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.
Here’s how the shell method can give you a solution:
Find an expression that represents the area of a random shell of the solid (in terms of x).
The Sum Rule for integration allows you to split a sum inside an integral into the sum of two separate integrals. Similarly, you can break a sum inside a series into the sum of two separate series:
For example:
A little algebra allows you to split this fraction into two terms:
Now the rule allows you to split this result into two series:
This sum of two series is equivalent to the series that you started with.
In calculus, 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:
The logarithmic function ln x
The first four inverse trig functions (arcsin x, arccos x, arctan x, and arccot x)
Beyond these cases, integration by parts is useful for integrating the product of more than one type of function or class of function.
Calculus
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. But as functions begin to get a little bit more complex, these methods become insufficient. For example, these methods don’t work on the following:
To evaluate this integral, you need some stronger medicine.
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