Section 4.3

1. Section 4.3 • In the 1800’s, the German mathematician, Georg Riemann, used the limit of a sum to define the area of a region in a plane.

2. Riemann Sum n = # of rectangles (partitions) f (xi) = height of each rectangle ∆xi = width of each rectangle

3. Consider the following limit: = L → Area under the curve

4. Definition of a Definite Integral • If f is defined on the closed interval [a, b] and • exists, then f is integrable (can be integrated) on [a, b] and the limit is denoted by This symbol means the sum from a to b.

5. Vocabulary

6. The limit is called the definite integral. This is always a number. • The number “a” is called the lower limit of integration. • The number “b” is called the upper limit of integration. • The function “f (x)” is called the integrand.

7. Area under a curve can be represented using a definite integral. f (x)

8. Examples

9. 8 Area of rectangle = L ∙ W = 2 ∙ 4

10. 4 Area of rt. ∆

11. 2 Area of semicircle

12. -3 Area I is a negative #. II Area II is a positive #. I

13. Properties of Definite Integrals

14. 0 No area under the curve

15. where a < b

16. Split the interval into parts like ex. 4.

18. Examples

20. HW: p. 278 (13-43 odd, 46, 47, 49)