1. RIEMANN SUMS AND DEFINITE INTEGRALS
Section 4.3

2. When you are done with your homework, you should be able to…
Understand and use the definition of a Riemann sum
Evaluate a definite integral using limits
Evaluate a definite integral using properties of definite integrals

3. Kepler lived from 1570-1640 in Poland
Kepler lived from in Poland. He discovered 3 laws of planetary motion: Orbits are ellipses, the time around the earth squared is directly proportional to the average distance to the sun cubed, and equal areas are swept out in equal time. Why are the 3 laws significant?
They confirmed the Pythagorean Philosophy of nature.
They confirmed the value of empiricism.
The led to the discovery of “dark matter”.
All of the above.

4. RIEMANN SUMS
Let f be defined on the closed interval and let be a partition of given by
where is the width of the ith subinterval. If is any point in the ith subinterval, then the
sum is called a
Riemann Sum of f for the partition .

5. NORM OF THE PARTITION
The width of the largest subinterval of a partition is the norm of the partition and is denoted by

6. REGULAR PARTITION
If every subinterval is of equal width, the partition is regular and the norm is denoted by

7. RELATIONSHIP BETWEEN THE NORM AND THE NUMBER OF SUBINTERVALS
For a general partition, the norm is related to the number of subintervals of in the following way:
So the number of subintervals in a partition approaches infinity as the norm of the partition approaches zero.

8. DEFINITE INTEGRALS Consider the following limit:
To say that this limit exists means that for every there exists a such that for every partition with it follows that

9. Definition of a Definite Integral
If f is defined on the closed interval and the
limit exists, then f is
integrable on and the limit is denoted by
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.

10. What is the difference between definite integrals and indefinite integrals?
A definite integral is a family of functions and an indefinite integral is a number.
A definite integral is a number and an indefinite integral is a family of functions.
There is no difference between a definite integral and an indefinite integral.

11. Evaluate the definite integral

12. Theorem: The definite Integral as the Area of a Region
If f is continuous and nonnegative on the closed interval , then the area of the region bounded by the graph of f, the x-axis, and the vertical lines and is given by

13. Good news, Bad news…
The good news is that in the next section we will be learning how to directly evaluate definite integrals.
The bad news is that for now you must use
The limit definition,
Or you can check to see whether the definite integral represents the area of a common geometric region such as a rectangle, triangle, or semicircle.

14. Definitions of Two Special Definite Integrals
If is defined at , then we define
If is integrable on , then we define

15. Theorem: Preservation of Inequality
If f is integrable and nonnegative on the closed interval , then
If f and g are integrable on the closed interval and for every x in , then

16. Additive Interval Property
If f is integrable on the three closed intervals determined by a, b, and c, then

17. Properties of Definite Integrals
If f and g are integrable on and k is a constant, then the functions of and are integrable on , and
and