# Section 6.4 - The Definite Integral ```Math 142 Lecture Notes for Section
Section 6.4 -
1
The Definite Integral
Definition 6.4.1:
If f is a function defined for a ≤ x ≤ b, we can divide the interval [a, b] into n subintervals
of equal width ∆x = (b − a)/n. We let x0 = a, xn = b and x∗1 , x∗2 , . . . , x∗n be any sample
points in the subintervals, so x∗i lies in the ith subinterval [xi−1 , xi ]. Then the definite
integral of f from a to b is
provided that this limit exists. If it does exist, we say that f is integrable on [a, b].
Example 6.4.2:
R3
Approximate 0 (2x2 − x − 2)dx by using the Riemann sum with 6 subintervals, taking
the sample points to be the midpoints of each subinterval.
Math 142 Lecture Notes for Section
Example 6.4.3:
(a)
R3
(b)
R4
(c)
R
1
2dx
0
|x − 2|dx
(
−3)3 4x + 5dx
2
Math 142 Lecture Notes for Section
3
Order Properties of Definite Integrals: Assume that all integrals given below exist
and that a ≤ b.
(1) If f (x) ≥ 0, then
(2) If f (x) ≤ g(x) on a ≤ x ≤ b, then
(3) if m ≤ f (x) ≤ M on a ≤ x ≤ b, then
Example 6.4.4:
Use the bounds of the function y = x3 to estimate
Z 4
x3 dx
0
Section 6.4 Suggested Homework 1-11 (odd, n = 10 only), 23-29(odd), 33-41(odd), 67
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