Section 5.2a
First, we need a reminder of sigma notation:
How do we evaluate:
n
a
k 1
k
 a1  a2  a3 
 an1  an
…and what happens if an “infinity” symbol appears
above the sigma???
 The terms go on indefinitely!!!
LRAM, MRAM, and RRAM are all examples of
Riemann sums, because of how they were constructed.
In this section, we start with a more general account
of these sums………………….observe…………….
We start with an arbitrary function f(x), defined on a closed
interval [a, b].
Partition the interval [a, b] into n subintervals by choosing n – 1
points between a and b, subject only to
a  x1  x2 
a
 xn1  b
b
Letting a = x0 and b = x n , we have a
partition of [a, b]:
P  x0 , x1 , x2 ,
, xn 
We start with an arbitrary function f(x), defined on a closed
interval [a, b].
The partition P determines n closed subintervals.
The
k th
subinterval is
x1
x0  a
 xk 1 , xk  , which has length
xk  xk  xk 1
x2
x1
xk
x2
xk 1
xn
xk
xn 1
xn  b
In each subinterval we choose some number, denoting the
th
number chosen from the k subinterval by ck .
On each subinterval, we create a rectangle that reaches from
the x-axis to touch the curve at ck , f  ck  .


x
On each subinterval, we create a rectangle that reaches from
the x-axis to touch the curve at ck , f  ck  .

c , f  c 
n
c , f  c 
k
c1

n
k
c2
ck
a
 c1, f  c1 
cn b
On each subinterval, we form the
product
f  ck  x
c , f  c 
2
2
(which can be positive,
negative, or zero…)
Area of each rectangle!!!
c , f  c 
n
c , f  c 
k
c1
k
c2
ck
a
c , f c 
1
n
1
cn b
Finally, take the sum of these products:
n
c , f  c 
2
2
S n   f  ck  xk
k 1
This sum is called the
Riemann sum for f on the interval [a, b]
Riemann Sums
As with LRAM, MRAM, and RRAM, all Riemann sums for a
given interval [a, b] will converge to common value, as long as
the subinterval lengths all tend to zero.
To ensure this last condition, we require that the longest
subinterval (called the norm of the partition, denoted ||P||)
tends to zero…
Definition: The Definite Integral as a
Limit of Riemann Sums
Let f be a function defined on a closed interval [a, b]. For any
partition P of [a, b], let numbers ck be chosen arbitrarily in the
subintervals x , x .

k 1
k

If there exists a number I such that
n
lim  f  ck  xk  I
P 0
k 1
no matter how P and the ck ‘s are chosen, then f is integrable
on [a, b] and I is the definite integral of f over [a, b].
Theorem: The Existence of Definite
Integrals
In particular, if f is continuous, then choices about partitions and
ck ‘s don’t matter, as long as the longest subinterval tends to
zero:
All continuous functions are integrable. That is, if a
function f is continuous on an interval [a, b], then its
definite integral over [a, b] exists.
This theorem allows for a simpler definition of the definite integral
for continuous functions. We need only consider the limit of
regular partitions (in which all subintervals have the same
length)…
The Definite Integral of a Continuous
Function on [a, b]
Let f be continuous on [a, b], and let [a, b] be partitioned into n
subintervals of equal length x  b  a n . Then the
definite integral of f over [a, b] is given by


n
lim  f  ck  x
n 
where each
k 1
ck is chosen arbitrarily in the k th subinterval.
Integral Notation
The Greek “S” is changed to an elongated Roman “S,”
so that the integral retains its identity as a “sum.”
n
b
k 1
a
lim  f  ck  x   f  x  dx
n 
This is read as “the integral from a to b of f of x dee x”
or “the integral from a to b of f of x with respect to x”
Integral Notation
The function is
the integrand
Upper limit
of integration
b
Integral
Sign
Lower limit
of integration

f  x  dx
x is the variable
of integration (also
called a dummy
variable)
a
Integral of f
from a to b
When you find the value
of the integral, you have
evaluated the integral
A Quick Practice Problem
The interval [–1, 3] is partitioned into n subintervals of equal
length x  4 n . Let mk denote the midpoint of the k th
subinterval. Express the given limit as an integral.
n


lim  3  mk   2mk  5 x
n 
k 1
2
The function being integrated is
over the interval [–1, 3]...
3
3
x


1
2
f  x   3x  2 x  5
 2 x  5  dx
2
Definition: Area Under a Curve (as a
Definite Integral
 
If y  f x is nonnegative and integrable over
a closed interval [a, b], then the area under
the curve of y  f x from a to b is the
integral of f from a to b,
 
b
A   f  x  dx
a
Practice Problem

Evaluate the integral
2
4  x dx
2
2
What is the graph of the integrand???
From Geometry-Land:
(0, 2)
1
1
2
2
 r    2   2
2
2
Area =
(–2, 0)
(2, 0)

2
2
4  x dx  2
2
What happens when the curve is
below the x-axis?
 The area is negative!!!
b
Area =
  f  x  dx
a
when
f  x  0
If an integrable function y = f (x) has both positive and negative
values on the interval [a, b], add the areas of the rectangles
above the x-axis, and subtract those below the x-axis:
For any integrable function,

b
a
f  x  dx
= (area above x-axis) – (area below x-axis)
What happens with constant
functions?
If f (x) = c, where c is a constant, on the
interval [a, b], then

b
a
f  x  dx  a  c  dx  c  b  a 
b
Does this make sense graphically???
Quick Example:

8
3


2

40
8
dx






2
3
Practice Problems
Use the graph of the integrand and areas to evaluate
the given integral.
32

2
x

4
dx


1 2
1
 1 3  1  2
2
Practice Problems
Use the graph of the integrand and areas to evaluate
the given integral.
1
1

x
dx



1
1
  2 1  1
2
Practice Problems
Use the graph of the integrand and areas to evaluate
the given integral.
b

3b
a
3tdt
1
  b  a  3b  3a 
2
3a
a
b
3 2
2
 b  a 
2