The Definite Integral
Believe it or not, we almost have the definition of the definite integral in hand. We will state it formally
so that we can refer to it conveniently as needed.
Definition: Let P be a partition of the interval [a, b], P = {x
0
, x
1
, x
2
, . . . , x
n
} with a = x
0
≤ x
1
≤ x
2
≤
··· ≤ x
n
= b. Let ∆x
i
= x
i
−x
i−1
be the width of the i
th
subinterval, 1 ≤ i ≤ n. Let f be a function defined
on [a, b]. Next form the Upper Riemann Sum U(P, f) where the height of the rectangle on each subinterval
is the maximum value of f on that subinterval; and form the Lower Riemann Sum L(P, f), where the height
of the rectangle on each subinterval is the minimum height of f on that subinterval. Then we say that f
is Riemann integrable on [a, b] if there exists a unique number Φ such that L(P, f) ≤ Φ ≤ U(P, f) for all
partitions of [a, b]. We write the number Φ as
Φ =
Z
b
a
f(x)dx
and call it the definite integral of f over [a, b].
The integral symbol is a stylized Greek sigma Σ from the summation notation we introduced above. The
x is a so-called dummy variable in that it merely tells us the variable with respect to which we are integrating;
hence, we could equally well write
R
b
a
f(t)dt or
R
b
a
f(r)dr.
The definition looks a bit awkward to verify. However, there are two important theorems that come to
our aid from advanced analysis, and which we rely on in practice.
Theorem 1: If f is Riemann integrable on [a, b], then
Z
b
a
f(x)dx = lim
n→∞
||P ||→0
n
X
i=1
f(c
i
)∆x
i
where c
i
is any point in the subinterval [x
i−1
, x
i
], and ||P || is the maximum length of the ∆x
i
.
So, the Upper Riemann Sum, the Lower Riemann Sum, the Left Riemann Sum, and the Right Riemann
Sum are all special cases of the sum in the above limit where we choose the points c
i
in very particular ways.
(That is, where f is a maximum, or a minimum, or the left endpoint, or the right endpoint, respectively.)
In the examples, we usually take the subintervals to be of equal length, so as n → ∞, the length of each
subinterval automatically goes to 0.
The above theorem is much more than a theoretical result. We will see that we use it extensively in
applications as a guide in setting up a mathematical model connected with the problem. But more about
that later. The theorem below allows us to work effectively with the integral because most of the functions
in which we will be interested are continuous or piecewise continuous.
Theorem 2: If f is continuous on [a, b], then f is Riemann integrable on [a, b].
This theorem tells us that for continuous functions, we can use the limit of any convenient Riemann sums
to evaluate the integral.
Example 7: Use an Upper Riemann Sum and a Lower Riemann Sum, first with 8, then with 100
subintervals of equal length to approximate the area under the graph of y = f (x) = x
2
on the interval [0, 1].
First with 8 subintervals:
U(P, f) =
1
8
8
X
i=1
i
2
64
≈ .3984375
L(P, f) =
1
8
7
X
i=0
i
2
64
≈ .2734375
7