4.3. THE DEFINITE INTEGRAL
237
function within a given subinterval, because
lim
n→∞
L n = lim
n→∞
R n = lim
n→∞
M n = lim
n→∞
n
i=1
f (x
∗
i )△x.
That these limits always exist (and share the same value) for a continuous 7 function f
allows us to make the following definition.
Definition 4.1. The definite integral of a continuous function f on the interval [a, b],
denoted
b
a
f (x) dx, is the real number given by
b
a
f (x) dx = lim
n→∞
n
i=1
f (x
∗
i )△x,
where △x =
b−a
n , x i = a + i△x (for i = 0, . . . , n), and x ∗
i satisfies x i−1 ≤ x ∗
i ≤ x i (for
i = 1, . . . , n).
We call the symbol
the integral sign, the values a and b the limits of integration, and
the function f the integrand. The process of determining the real number
b
a
f (x) dx is
called evaluating the definite integral. While we will come to understand that there are
several different interpretations of the value of the definite integral, for now the most
important is that
b
a
f (x) dx measures the net signed area bounded by y = f (x) and the
x-axis on the interval [a, b]. For example, in the notation of the definite integral, if f is
the function pictured in Figure 4.22 and A 1 , A 2 , and A 3 are the exact areas bounded by f
and the x-axis on the respective intervals [a, b], [b, c], and [c, d], then
b
a
f (x) dx = A 1 ,
c
b
f (x) dx = −A 2 ,
d
c
f (x) dx = A 3 ,
and
d
a
f (x) dx = A 1 − A 2 + A 3 .
We can also use definite integrals to express the change in position and distance traveled
by a moving object. In the setting of a velocity function v on an interval [a, b], it follows
from our work above and in preceding sections that the change in position, s(b) − s(a), is
given by
s(b) − s(a) =
b
a
v(t) dt.
7 It turns out that a function need not be continuous in order to have a definite integral. For our purposes,
we assume that the functions we consider are continuous on the interval(s) of interest. It is straightforward
to see that any function that is piecewise continuous on an interval of interest will also have a well-defined
definite integral.
237
function within a given subinterval, because
lim
n→∞
L n = lim
n→∞
R n = lim
n→∞
M n = lim
n→∞
n
i=1
f (x
∗
i )△x.
That these limits always exist (and share the same value) for a continuous 7 function f
allows us to make the following definition.
Definition 4.1. The definite integral of a continuous function f on the interval [a, b],
denoted
b
a
f (x) dx, is the real number given by
b
a
f (x) dx = lim
n→∞
n
i=1
f (x
∗
i )△x,
where △x =
b−a
n , x i = a + i△x (for i = 0, . . . , n), and x ∗
i satisfies x i−1 ≤ x ∗
i ≤ x i (for
i = 1, . . . , n).
We call the symbol
the integral sign, the values a and b the limits of integration, and
the function f the integrand. The process of determining the real number
b
a
f (x) dx is
called evaluating the definite integral. While we will come to understand that there are
several different interpretations of the value of the definite integral, for now the most
important is that
b
a
f (x) dx measures the net signed area bounded by y = f (x) and the
x-axis on the interval [a, b]. For example, in the notation of the definite integral, if f is
the function pictured in Figure 4.22 and A 1 , A 2 , and A 3 are the exact areas bounded by f
and the x-axis on the respective intervals [a, b], [b, c], and [c, d], then
b
a
f (x) dx = A 1 ,
c
b
f (x) dx = −A 2 ,
d
c
f (x) dx = A 3 ,
and
d
a
f (x) dx = A 1 − A 2 + A 3 .
We can also use definite integrals to express the change in position and distance traveled
by a moving object. In the setting of a velocity function v on an interval [a, b], it follows
from our work above and in preceding sections that the change in position, s(b) − s(a), is
given by
s(b) − s(a) =
b
a
v(t) dt.
7 It turns out that a function need not be continuous in order to have a definite integral. For our purposes,
we assume that the functions we consider are continuous on the interval(s) of interest. It is straightforward
to see that any function that is piecewise continuous on an interval of interest will also have a well-defined
definite integral.