Section 6.3: The Definite Integral
Definition: If f is a function defined on a closed interval [a, b], let P be a partition of [a, b]
with partition points x0 , x1 , x2 , . . . , xn , where
a = x0 < x1 < x2 < · · · < xn = b.
Choose representative points x∗i in [xi−1 , xi ] and let ∆xi = xi − xi−1 and ||P || = max{∆xi }.
Then the definite integral of f from a to b is
Z
b
f (x)dx = lim
||P ||→0
a
n
X
f (x∗i )∆xi ,
i=1
provided that this limit exists. If the limit does exist, then f is called integrable on [a, b].
The function f (x) is called the integrand and a and b are called the limits of integration.
Theorem: (Midpoint Rule)
Any Riemann sum is an approximation to an integral. If we use a regular partition and
midpoints as the representative points, then
Z
b
f (x)dx ≈
a
where ∆x =
n
X
f (x̄i )∆x
i=1
xi−1 − xi
b−a
and x̄i =
.
n
2
Example: Use the Midpoint Rule with n = 4 rectangles to approximate the definite integral
Z
2
√
1 + x2 dx.
1
1
Note: If f (x) ≥ 0, then the definite integral of f from a to b represents the area under the
graph of y = f (x) on [a, b].
Example: Evaluate each integral by interpreting it in terms of areas.
Z 2√
(a)
4 − x2 dx
−2
Z
3
(2 − x)dx
(b)
−1
Z
3
|3x − 6|dx
(c)
0
2
Note: To evaluate a definite integral as the limit of a Riemann sum, it is often easier to use a
regular partition. That is, a partition for which all the subintervals have the same length,
b−a
.
n
∆x =
If we choose x∗i to be the right endpoint, then
x∗i
= a + i∆x = a +
b−a
n
i.
Then the definite integral of f from a to b is
Z
b
f (x)dx = lim
a
Z
Example: Evaluate
n→∞
n
X
f (x∗i )∆x.
i=1
5
(2 + 3x − x2 )dx as the limit of a Riemann sum.
1
3
Example: Express each limit as a definite integral.
n
X
i4
(a) lim
n→∞
n5
i=1
n
1
1X
n→∞ n
1 + (i/n)2
i=1
(b) lim
" #
5
n
2X
2i
(c) lim
3 1+
−6
n→∞ n
n
i=1
4
Theorem: (Properties of the Definite Integral)
Suppose that f and g are integrable functions and c is a constant.
Z b
cdx = c(b − a)
1.
a
Z
b
2.
b
Z
cf (x)dx = c
f (x)dx
a
Z
a
b
Z
Z
g(x)dx
a
a
a
b
Z
f (x)dx ±
[f (x) ± g(x)]dx =
3.
b
a
f (x)dx = 0
4.
a
Z
b
a
Z
f (x)dx = −
5.
f (x)dx
a
Z
6.
b
b
c
Z
f (x)dx =
a
Z
f (x)dx +
a
b
f (x)dx
c
7. If f (x) ≤ g(x) for a ≤ x ≤ b, then
Z
b
Z
f (x)dx ≤
b
g(x)dx.
a
a
8. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then
Z
b
f (x)dx ≤ M (b − a).
m(b − a) ≤
a
Z
Example: Evaluate
1
x2 cos xdx.
1
5
Example: Write each expression as a single integral.
Z 7
Z 10
f (x)dx
f (x)dx −
(a)
2
2
Z
5
Z
0
f (x)dx −
(b)
−3
Z
f (x)dx +
−3
6
f (x)dx
5
Z
Example: Find an upper and lower bound for
0
6
2
√
x3 + 1dx.