This document discusses Riemann sums and the definite integral. It explains that the definite integral is defined as the limit of Riemann sums as the size of the subintervals approaches zero. It provides examples of calculating Riemann sums and shows how the definite integral can be approximated by Riemann sums. The document also outlines some key properties of the definite integral, such as how to integrate sums and how the integral relates to calculating the area under a curve.

1. Riemann Sums and the
Definite Integral

2. Why?
Why is the area of the yellow rectangle at the end (look
at the applet on this lesson)
 =
a bx∆
( )( ) ( )x f b f a∆ × −

3. Review
We partition the interval into n sub-intervals
Evaluate f(x) at right endpoints
of kth
sub-interval for k = 1, 2, 3, … n
a b
f(x)
b a
x
n
−
∆ =
a k x+ ×∆

4. Review
Sum
We expect Sn to improve
thus we define A, the area under the curve, to equal
the above limit.
a b
1
lim ( )
n
n
n
k
S f a k x x
→∞
=
= + ×∆ ×∆∑
f(x)
Look at
Goegebra demo
Look at
Goegebra demo

5. Riemann Sum
1. Partition the interval [a,b] into n subintervals
a = x0 < x1 … < xn-1< xn = b
1. Call this partition P
2. The kth
subinterval is ∆xk = xk-1 – xk
3. Largest ∆xk is called the norm, called ||P||
— Choose an arbitrary value from each
subinterval, call it
ic

6. Riemann Sum
3. Form the sum
4. This is the Riemann sum associated with
• the function f
• the given partition P
• the chosen subinterval representatives
 We will express a variety of quantities in terms of the
Riemann sum
1 1 2 2
1
( ) ( ) ... ( ) ( )
n
n n n i i
i
R f c x f c x f c x f c x
=
= ∆ + ∆ + + ∆ = ∆∑
ic

7. The Riemann Sum
Calculated
Consider the function
2x2
– 7x + 5
Use ∆x = 0.1
Let the = left edge
of each subinterval
Note the sum
x 2x^2-7x+5 dx * f(x)
4 9 0.9
4.1 9.92 0.992
4.2 10.88 1.088
4.3 11.88 1.188
4.4 12.92 1.292
4.5 14 1.4
4.6 15.12 1.512
4.7 16.28 1.628
4.8 17.48 1.748
4.9 18.72 1.872
5 20 2
5.1 21.32 2.132
5.2 22.68 2.268
5.3 24.08 2.408
5.4 25.52 2.552
5.5 27 2.7
5.6 28.52 2.852
5.7 30.08 3.008
5.8 31.68 3.168
5.9 33.32 3.332
Riemann sum = 40.04
ic

8. The Riemann Sum
We have summed a series of boxes
If the ∆x were smaller, we would have gotten a better
approximation
f(x) = 2x2
– 7x + 5
1
( ) 40.04
n
i i
i
f c x
=
∆ =∑

9. The Definite Integral
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
( )0
1
lim( )
b
a P
n
i i
k
f f c xI x dx
→
=
= ∆= ∫ ∑

10. If f is defined on the closed interval [a, b] and the limit of
a Riemann sum of f exists, then we say f is integrable on [a, b]
and we denote the limit by
∫∑ =∆
=
→∆
b
a
n
i
ii
x
dxxfxcf )()(lim
1
0
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.
Definition of the Definite
Integral

11. Example
Try
Use summation on calculator.
( )
3 4
2
4
11
use (1 )
k
x dx S f k x x
=
= + ⋅∆ ∆∑∫
b a
x
n
−
∆ =

12. Example
Note increased accuracy with smaller ∆x

13. Limit of the Riemann Sum
The definite integral is the
limit of the Riemann sum.
( )
3
2
1
x dx∫

14. Properties of Definite Integral
Integral of a sum = sum of integrals
Factor out a constant
Dominance
( ) ( ) [ , ]
( ) ( )
b b
a a
f x g x on a b
f x dx g x dx
≤
≤∫ ∫

15. Properties of Definite Integral
Subdivision rule
( ) ( ) ( )
c b c
a a b
f x dx f x dx f x dx= +∫ ∫ ∫
a b c
f(x)

16. Area As An Integral
The area under
the curve on the
interval [a,b]
a c
f(x)
( )
b
a
A f x dx= ∫
A