Section 4.3 – Riemann Sums
and Definite Integrals
Riemann Sums
The rectangles need not have equal width, and the
height may be any value of f(x) within the subinterval.
1. Partition (divide) [a,b]
into N subintervals.
2. Find the length of
each interval:
xi  xi  xi 1
c1
a=x0 x1
c2
ci
c3
x2
x3
 f  c  x
i 1
i
cN
xi
N
i
3. Find any point ci in
the interval [xi,xi-1].
b =xN
4. Construct every
rectangle of height f(ci)
and base Δxi.
4. Find the sum of the
areas.
Riemann Sums
The norm of P, denoted ││P││, is the maximum of
the lengths Δxi.
a
b
As ││P││ gets closer to 0, the sum of the areas of the rectangles is
closer to the actual area under the curve
Riemann Sums
The norm of P, denoted ││P││, is the maximum of
the lengths Δxi.
a
b
As ││P││ gets closer to 0, the sum of the areas of the rectangles is
closer to the actual area under the curve
Definite Integral
The definite integral of f(x) over [a,b], denoted by the
integral sign, is the limit of Riemann sums:
Upper limit of
integration

b
a
N
f  x  dx  lim  f  ci  xi
P 0
i 1
Lower limit of
integration
Where the limit exists, we say that f(x) is integrable
over [a,b].
Notation Examples
The definite integral that represents the area is…
EX1:
f(x)
S   f  x  dx
b
a
S
a
b
Ex2: The area under the
parabola y=x2 from 0 to 1
1
A   x dx
0
2
Theorem: The Existence of Definite
Integrals
If f(x) is continuous on [a,b], or if f(x) is continuous with
at most finitely many jump discontinuities (one sided
limits are finite but not equal), then f(x) is integrable
over [a,b].
a
b
Negative Area or “Signed” Area
If a function is less than zero for an interval,
the region between the graph and the xaxis represents negative area.
Positive Area
Negative Area
Definite Integral: Area Under a Curve
If y=f(x) is integrable over a closed interval
[a,b], then the area under the curve y=f(x)
from a to b is the integral of f from a to b.
Upper limit of
integration
 f  x  dx   area above the x-axis    area below the x-axis 
b
a
Lower limit of
integration
Example 1
  4  x  dx .
9
Calculate
1
  4  x  dx
9
1
   4  x  dx    4  x  dx
4
1
9
4
  12  3  3    12  5  5 
 8
Example 2
Calculate

9
1
4  x dx .

9
1
4  x dx
4
9
  4  x dx   4  x dx
1
4
  12  3  3   12  5  5 
 17
Rules for Definite Integrals
Let f and g be functions and x a variable; a, b, c, and k be constant.

Constant
a

Constant Multiple
Sum Rule
b
a

b
a
k dx  k  b  a 
kf  x  dx  k  f  x  dx
b
a
 f  x   g  x   dx   f  x  dx   g  x  dx
a
a
b
b
 f  x  dx   f  x  dx
Reversing the Limits
Additivity
b
a
b
b
a
 f  x  dx   f  x  dx   f  x  dx
b
c
c
a
b
a
Example 1
If

b
0
x dx 
  2x
3
0
2
2
3
b
3
 5  dx 
, calculate

3
0
  2x
3
2
0
3
2 x dx   5 dx
2
Sum Rule
0
3
3
 2  x dx   5 dx
0
2
0
 2   5 3  0
33
3
3
 5  dx .
Constant Multiple
Rule
Given and
Constant Rule
Example 2

If
b
0

7

7
0
4
x dx  b2 , calculate
2
4

7
4
x dx .
7
x dx   x dx   x dx
0
4
7
4
0
0
Additivity Rule
x dx   x dx   x dx

72
2

 16.5
42
2
Given