riemann Sum

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