1. Riemann Sums & Definite Integrals

2. Finding Area with Riemann Sums
Subintervals with equal width
For convenience, the area of a partition is often divided into subintervals with equal width – in other words, the rectangles all have the same width. (see the diagram to the right and section 5-2)

3. Finding Area with Riemann Sums
It is possible to divide a region into different sized rectangles based on an algorithm or rule (see graph above and example #1 on page 307)

4. Finding Area with Riemann Sums
It is also possible to make rectangles of whatever width you want where the width and/or places where to take the height does not follow any particular pattern.
Notice that the subintervals don’t seem to have a pattern. They don’t have to be any specific width or follow any particular pattern.
Also notice that the height can be taken anywhere on each subinterval – not only at endpoints or midpoints!

5. What is a Riemann Sum Definition:
Let f be defined on the closed interval [a,b], and let  be a partition [a,b] given by a = x0 < x1 < x2 < … < xn-1 < xn = b where xi is the width of the ith subinterval [xi-1, xi]. If c is any point in the ith subinterval, then the sum
is called a Riemann sum of f for the partition .

6. New Notation for x b - a n
When the partitions (boundaries that tell you where to find the area) are divided into subintervals with different widths, the width of the largest subinterval of a partition is the NORM of the partition and is denoted by ||||
If every subinterval is of equal width, the partition is REGULAR and the norm is denoted by ||||= x =
The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. In other words, |||| implies that n 
b - a n
Is the converse of this statement true? Why or why not?

7. Definite Integrals
If f is defined on the closed interval [a,b] and the limit
exists, then f is integrable on [a,b] 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.

8. Definite Integrals vs. Indefinite Integrals
A definite integral is number.
An indefinite integral is a family of functions.
They may look a lot alike, however,
definite integrals have limits of integration while the
indefinite integrals have not limits of integration.
Definite Integral
Indefinite Integral

9. Theorem 5.4 Continuity Implies Integrability
If a function f is continuous on the closed interval [a,b], then f is integrable on [a,b].
Is the converse of this statement true? Why or why not?

10. Evaluating a definite integral…
To learn how to evaluate a definite integral as a limit, study Example #2 on p. 310.

11. Theorem 5.5 The Definite Integral as the Area of a Region
If f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f , the x-axis, and the vertical lines x = a and x = b is given by
Area =

12. Let’s try this out… Area = (base)(height) = (2)(4) = 8 un 2
Sketch the region
Find the area indicated by the integral.
Area
= (base)(height)
= (2)(4) = 8 un 2

13. Give this one a try… Area of a trapezoid = .5(width)(base1+ base2)
Sketch the region
Find the area indicated by the integral.
base2 =5
Area of a trapezoid
= .5(width)(base1+ base2)
= (.5)(3)(2+5) = 10.5 un 2
base1=2
Width =3

14. Try this one… Area of a semicircle = .5( r2) = (.5)()(22) = 2  un 2
Sketch the region
Find the area indicated by the integral.
Area of a semicircle
= .5( r2)
= (.5)()(22)
= 2  un 2

15. Properties of Definite Integrals
If f is defined at x = a, then we define
So,
If f is integrable on [a,b], then we define
So,

16. Additive Interval Property
Theorem 5.6
Additive Interval Property
If f is integrable on three closed intervals determined by a, b, and c, then

17. Properties of Definite Integrals
Theorem 5.7
Properties of Definite Integrals
If f and g are integrable on [a,b] and k is a constant, then the functions of kf and f  g are integrable on [a,b], and

18. HOMEWORK – yep…more practice
Thursday, January 17
Read and take notes on section 5.3 and do p # 3, 6, 9, …, 45, 47, 49, 53, – Work on the AP Review Diagnostic Tests if you have time over the long weekend.
Tuesday – January 22 –
Get the Riemann sum program for your calculator and do p. 316 # 59-64, 71 and then READ and TAKE Notes on section 5-4 and maybe more to come!!