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CHAPTER 5
5
Computing area under a curve, like the area of the region spanned by the
scaffolding under the roller coaster track, is an application of integration.
INTEGRATION
1 Antidifferentiation: The Indefinite Integral
2 Integration by Substitution
3 The Definite Integral and the Fundamental Theorem of
Calculus
4 Applying Definite Integration: Area Between Curves and
Average Value
5 Additional Applications to Business and Economics
6 Additional Applications to the Life and Social Sciences
Chapter Summary
Important Terms, Symbols, and Formulas
Checkup for Chapter 5
Review Exercises
Explore! Update
Think About It
371
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372 CHAPTER 5 Integration 5-2
SECTION 5.1 Antidifferentiation: The Indefinite Integral
How can a known rate of inflation be used to determine future prices? What is the
velocity of an object moving along a straight line with known acceleration? How can
knowing the rate at which a population is changing be used to predict future popula-
tion levels? In all these situations, the derivative (rate of change) of a quantity is
known and the quantity itself is required. Here is the terminology we will use in con-
nection with obtaining a function from its derivative.
Antidifferentiation ■ Afunction F(x) is said to be an antiderivative of f(x)if
F(x) f(x)
for every x in the domain of f(x). The process of finding antiderivatives is called
antidifferentiation or indefinite integration.
NOTE Sometimes we write the equation
F(x) f(x)
as
dF f(x) ■
dx
Later in this section, you will learn techniques you can use to find antideriva-
tives. Once you have found what you believe to be an antiderivative of a function,
you can always check your answer by differentiating. You should get the original func-
tion back. Here is an example.
EXAMPLE 5.1.1
1 3 2
Verify that F(x) 3x 5x 2 is an antiderivative of f(x) x 5.
Solution
F(x) is an antiderivative of f(x) if and only if F(x) f(x). Differentiate F and you
will find that
1 2
F(x) 3(3x ) 5
x25f(x)
as required.
The General Afunction has more than one antiderivative. For example, one antiderivative of the
2 3
Antiderivative function f(x) 3x is F(x) x , since
of a Function 2
F(x) 3x f(x)
3 3 3
but so are x 12 and x 5 and x , since
d 3 2 d 3 2 d 3 2
dx(x 12) 3x dx(x 5) 3x dx(x ) 3x
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5-3 SECTION 5.1 ANTIDIFFERENTIATION: THE INDEFINITE INTEGRAL 373
In general, if F is one antiderivative of f, then so is any function of the form
G(x) F(x)C, for constant C since
G(x) [F(x) C]
F(x)C sum rule for derivatives
F(x)0 derivative of a constant is 0
f(x) since F is an antiderivative of f
Conversely, it can be shown that if F and G are both antiderivatives of f, then
G(x) F(x)C, for some constant C (Exercise 64). To summarize:
Fundamental Property of Antiderivatives ■ If F(x) is an antideriva-
tive of the continuous function f(x), then any other antiderivative of f(x) has the
form G(x) F(x) C for some constant C.
There is a simple geometric interpretation for the fundamental property of anti-
derivatives. If F and G are both antiderivatives of f, then
Just-In-Time REVIEW G(x) F(x) f(x)
This means that the slope F(x) of the tangent line to y F(x) at the point (x, F(x))
Recall that two lines are is the same as the slope G(x) of the tangent line to y G(x) at (x, G(x)). Since the
parallel if and only if their slopes are equal, it follows that the tangent lines at (x, F(x)) and (x, G(x)) are paral-
slopes are equal. lel, as shown in Figure 5.1a. Since this is true for all x, the entire curve y G(x)
must be parallel to the curve y F(x), so that
y G(x)F(x)C
In general, the collection of graphs of all antiderivatives of a given function f is a
family of parallel curves that are vertical translations of one another. This is illus-
trated in Figure 5.1b for the family of antiderivatives of f(x) 3x2.
EXPLORE!
Store the function F(x) x3 y G(x) y 3
into Y1 of the equation editor y x
in a bold graphing style.
Generate a family of vertical y F(x)
transformations Y2 Y1 L1, 3
y x
where L1 is a list of constants, (x, G(x))
{4, 2, 2, 4}. Use the
graphing window [4.7, 4.7]1 x
by [6, 6]1. What do you (x, F(x))
observe about the slopes of 3 5
all these curves at x 1? y x
x
(a) If F(x) G(x), the tangent lines at (b) Graphs of some members of the family
(x, F(x)) and (x, G(x)) are parallel of antiderivatives of f(x) 3x2
FIGURE 5.1 Graphs of antiderivatives of a function f form a family of parallel curves.
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374 CHAPTER 5 Integration 5-4
The Indefinite You have just seen that if F(x) is one antiderivative of the continuous function f(x),
Integral then all such antiderivatives may be characterized by F(x) C for constant C. The
family of all antiderivatives of f(x) is written
f(x) dx F(x) C
and is called the indefinite integral of f(x). The integral is “indefinite” because it
involves a constant C that can take on any value. In Section 5.3, we introduce a
definite integral that has a specific numerical value and is used to represent a vari-
ety of quantities, such as area, average value, present value of an income flow, and
cardiac output, to name a few. The connection between definite and indefinite inte-
grals is made in Section 5.3 through a result so important that it is referred to as the
Just-In-Time REVIEW fundamental theorem of calculus.
In the context of the indefinite integral f(x) dx F(x) C, the integral symbol
Recall that differentials were is , the function f(x) is called the integrand, C is the constant of integration, and
introduced in Section 2.5. dx is a differential that specifies x as the variable of integration. These features are
2
displayed in this diagram for the indefinite integral of f(x) 3x :
integrand ⎯⎯⎯⎢ ⎢ ⎯constant of integration
↓ ↓
3x2dx x3 C
↓ ⎢↓
⎢ ⎯⎯⎯⎯⎯⎯
integral symbol⎯⎯ variable of integration
For any differentiable function F, we have
F(x) dx F(x) C
EXPLORE!
Most graphing calculators since by definition, F(x) is an antiderivative of F(x). Equivalently,
allow the construction of
an antiderivative through dF dx F(x) C
its numerical integral, dx
fnInt(expression, variable,
lower limit, upper limit), found This property of indefinite integrals is especially useful in applied problems where a
in the MATH menu. In the rate of change F(x) is given and we wish to find F(x). Several such problems are
equation editor of your examined later in this section, in Examples 5.1.4 through 5.1.8.
calculator write
Y1fnInt(2X, X, {0, 1, 2}, X) It is useful to remember that if you have performed an indefinite integration
and graph using an expanded calculation that leads you to believe that f(x) dx G(x) C, then you can check
decimal window, [4.7, 4.7]1 your calculation by differentiating G(x):
by [5, 5]1. What do you If G(x) f(x), then the integration f(x) dx G(x) C is correct, but if
observe and what is the G(x) is anything other than f(x), you’ve made a mistake.
general form for this family
of antiderivatives? This relationship between differentiation and antidifferentiation enables us to estab-
lish these integration rules by “reversing” analogous differentiation rules.
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