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we need to take it to the limit one more time.
34
By taking the limit as n
increases without bound and the slices become infinitesimally small, we get
our definition of the definite integral.
Definite Integral
n
b
f(x) dx = lim
n→∞
f(x
i
)∆x.
(11.22)
a
i=1
In this equation,
∆x = (b − a)/n,
x
i
= a + i∆x.
Equation (11.22) is read as “The integral from a to b of f(x) dx.” Some
people read dx as “with respect to x.” The great similarity in notation
between the left- and right-hand sides of Equation (11.22) is by design. Just
like with the derivative, the finite step size ∆x becomes the infinitesimal
dx. The sigma symbol
used for discrete summations is replaced with
the symbol
, which is an elongated S that Leibniz intended to stand for
“summation.”
35
The a and b are known as the “limits of integration” and
define the starting and ending points. The function being integrated is
called the integrand.
An integral defined as a sum of “vertical slices” like this is known as a
Riemann integral. It's the most common definition, but not the most gen-
eral. In fact, our definition is not quite as general as the typical definition
of a Riemann integral. The astute reader may notice that ∆x is a constant,
and could be pulled in front of the summation, making it ∆x
n
i=1
f(x
i
).
That works in this case because we are using a regular partition, and all the
slices are the same width. In general, however, this restriction is not neces-
sary. The traditional definition of the Riemann integral takes the limit as
the width of the largest slice goes to zero. Our definition is certainly pow-
erful enough for well-behaved functions we deal with, but more powerful
definitions are needed to integrate more esoteric functions. Furthermore,
34
A shout out to all the Eagles fans out there who got that joke. The rest of you will
find out in the long run. Just take it easy, get over it, and you'll get a peaceful, easy
feeling.
35
Actually, “summierung” since he spoke German; we're just lucky it works in English,
too.
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