Chapter 4 notes from the Text Book Section 4.3 Riemann Sums and Definite Integrals 
Definition of Riemann
Sum Let f be defined on the closed interval [a,b], and let Δ
be a partition of [a,b] given by a
= x_{0} < x_{1} < x_{2} < x_{3}
< . . . . < x_{n1} < x_{n} = b where Δx_{ί }is
the width of the ίth subinterval.
If C_{ ί }is any point in the ίth subinterval, the then sum
( when the upper boundary is n and the lower boundary is ί=1) Σ f(C_{ }_{ί }) Δx_{ί, }x_{n1 }≤ C_{ }_{ί } ≤ x_{ί, } is called a Riemann sum of f for the partition Δ.

 Δ
 = Δx_{ }= (ba) / n
regular partition

Definition of a
Definite Integral If f is defined on the closed interval [a,b] and the limit lim( Δ
 à
0) Σ
f(C_{ }_{ί
}) Δx_{ί
}=_{
}∫ _{a}^{ b}^{ }f(x)
dx The limit is called the
definite integral of f from a to b. The
number a is the lower limit of integration, and the number b is the upper
limit of integration. 
Theorem 4.4
Continuity Implies Integrability If a function f is
continuous on the closed interval [a,b], then f is integrable on the [a,b].

Theorem 4.5
The Definite Integral as the Area of a Region If f is continuous and nonnegative on the close interval [a,b],
then the area of the region bounded by the graph of f, the xaxis, and the
vertical lines x = a
Area =_{ }∫ _{a}^{ b}^{ }f(x)
dx 
Example 6
Evaluation of a Definite Integral Evaluate ∫ _{1}^{
3 }(x^2 + 4x 3) dx using
each of the fallowing values. ∫ _{1}^{ 3
}(x^2) dx = 26/3, ∫
_{1}^{ 3 }(x) dx = 4, ∫ _{1}^{ 3 } dx = 2 Solution ∫ _{1}^{ 3
}(x^2 + 4x 3) dx =
∫ _{1}^{ 3 }(x^2) dx
+ ∫ _{1}^{ 3 }(4x ) dx + ∫ _{1}^{
3 }(3) dx
= ∫ _{1}^{ 3 }(x^2) dx
+ 4∫ _{1}^{ 3 } dx
 3 ∫ _{1}^{ 3 } dx
= (26/3) + 4(4) 3(2)
=
(4/3)
