1. 4.3 Riemann Sums and Definite Integrals

2. The Definite Integral
In the Section 4.2, the definition of area is defined as

3. The Definite Integral
The following example shows that it is not necessary to have subintervals of equal width
Example Find the area bounded by the graph of
and x-axis over the interval [0, 1].
Solution
Let ( i = 1, 2, …, n) be the endpoint of the subinteravls. Then the width of the i th subinterval is
The width of all subintervals varies.
Let ( i = 1, 2, …, n) be the point in the i th subinteravls, then

4. Continued… Example 1 Find the area bounded by the graph of
and x-axis over the interval [0, 1].
Solution
Let and ( i = 1, 2, …, n) be the endpoint of the subinteravls and the point in the i th subinterval.
So, the limit of sum is

5. Definition of a Riemann Sum

6. The Definite Integral = length of the i th subinterval a b
= partition of [a, b]
= length of the i th subinterval a
Norm of b
= length of the longest subinterval
Upper Limit
definition
area, f(x) > 0 on [a, b]
Lower Limit
net area, otherwise
Riemann Sum - approximates the definite integral
“definite integral of f from a to b”

7. variable of integration
upper limit of integration
Integration
Symbol
integrand
variable of integration
(dummy variable)
lower limit of integration
It is called a dummy variable because the answer does not depend on the variable chosen.

8. Definition of a Definite Integral

9. Theorem 4.4 Continuity Implies Integrability
Questions
Is the converse of Theorem 4.4 true? Why?
If change the condition of Theorem 4.4 “f is continuous” to “f is differentiable”, is the Theorem 4.4 true?
Of the conditions “continuity”, “differentiability” and “integrability”, which one is the strongest?

10. About Theorem 4.4 Continuity Implies Integrability
Answers
False. Counterexample is
Yes. Because “f is differentiable” implies “f is continuous”
The order from strongest to weakest is “integrability”, “continuity”, and “differentiability”.
1, when x ≠ 1 on [0, 5]
0, otherwise

11. The Definite Integral f A a b A1 f A3
= area above – area below a b A2

12. Special Cases
If using subintervals of equal length, (regular partition), with ci chosen as the right endpoint of the i th subinterval, then
Regular Right-Endpoint Formula (RR-EF)

13. Special Cases
If using subintervals of equal length, (regular partition), with ci chosen as the left endpoint of the i th subinterval, then
Regular Left-Endpoint Formula (RL-EF)

14. Theorem 4.6 Properties of the Definite Integralc b
by definition
by definition

15. Theorem 4.6 Properties of the Definite Integral1.
Reversing the limits changes the sign. 2.
If the upper and lower limits are equal, then the integral is zero. 3.
Constant multiples can be moved outside. 4.
Integrals can be added and subtracted.

16. Theorem 4.7 Properties of the Definite Integral

17. Examples
Example 2 If
and
then find
Solution

18. Homework
Pg , odd, odd, odd, 45-49, 55