Riemann Sums and the
Definite Integral
Lesson 5.3
Why?
• Why is the area of the yellow rectangle at the
end = x   f (b)  f (a) 
a
x
b
Review
f(x)
a
b
• We partition the interval into n sub-intervals
ba
x 
n
• Evaluate f(x) at right endpoints a  k x
of kth sub-interval for k = 1, 2, 3, … n
Review
f(x)
a
b
Look at
Goegebra demo
n
f (a  k  x)  x

• Sum S n  lim
n 
k 1
• We expect Sn to improve
thus we define A, the area under the
curve, to equal the above limit.
Riemann Sum
1. Partition the interval [a,b] into n subintervals
a = x0 < x1 … < xn-1< xn = b
•
•
•
Call this partition P
The kth subinterval is xk = xk-1 – xk
Largest xk is called the norm, called ||P||
2. Choose an arbitrary value from each
subinterval,
call it
ci
Riemann Sum
3. Form the sum
n
Rn  f (c1 )x1  f (c2 )x2  ...  f (cn )xn   f (ci )xi
i 1
This is the Riemann sum associated with
•
•
•
•
the function f
the given partition P
the chosen subinterval representatives
ci
We will express a variety of quantities in
terms of the Riemann sum
The Riemann Sum
Calculated
• Consider the function
2x2 – 7x + 5
• Use x = 0.1
• Let the ci = left edge
of each subinterval
• Note the sum
x
2x^2-7x+5
9
9.92
10.88
11.88
12.92
14
15.12
16.28
17.48
18.72
20
21.32
22.68
24.08
25.52
27
28.52
30.08
31.68
33.32
dx * f(x)
0.9
0.992
1.088
1.188
1.292
1.4
1.512
1.628
1.748
1.872
2
2.132
2.268
2.408
2.552
2.7
2.852
3.008
3.168
3.332
Riemann sum =
40.04
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
The Riemann Sum
f(x) = 2x2 – 7x + 5
n
 f (c )x
i 1
i
i
 40.04
• We have summed a series of boxes
• If the x were smaller, we would have gotten
a better approximation
The Definite Integral
b
I   f ( x)dx 
a
n
lim  f  ci  xi
P 0
k 1
• The definite integral is the limit of the
Riemann sum
• We say that f is integrable when
 the number I can be approximated as accurate
as needed by making ||P|| sufficiently small
 f must exist on [a,b] and the Riemann sum
must exist
Example
• Try
3
2
x
  dx
1
4
use S4   f (1  k  x)x
k 1
• Use summation on calculator.
ba
x 
n
Example
• Note increased accuracy with smaller x
Limit of the Riemann Sum
3
• The definite integral   x dx
1
limit of the Riemann sum.
2
is the
Properties of Definite Integral
• Integral of a sum = sum of integrals
• Factor out a constant
• Dominance
f ( x)  g ( x) on [a, b]
b

a
b
f ( x)dx   g ( x)dx
a
Properties of Definite Integral
f(x)
• Subdivision rule
a
b
c
b
c
a
a
b
 f ( x)dx   f ( x)dx   f ( x)dx
c
Area As An Integral
• The area under
the curve on the
interval [a,b]
b
f(x)
A
a
A   f ( x)dx
a
c
Distance As An Integral
• Given that v(t) = the velocity function
with respect to time:
• Then Distance traveled can be
determined by a definite integral
t b
D

v(t ) dt
t a
• Think of a summation for many small
time slices of distance
Assignment
• Section 5.3
• Page 314
• Problems: 3 – 47 odd