1. Riemann sums & definite integrals (4.3) January 28th, 2015

2. I. Riemann Sums Def. of a Riemann Sum: Let f be defined on the closed interval [a, b], and let be a partition of [a, b] given by where is the width of the ith subinterval. If is any point in the ith subinterval, then the sum is called the Riemann Sum of f for the partition.

3. *The norm of the partition is the largest subinterval and is denoted by. If the partition is regular (all the subintervals are of equal width), the norm is given by.

4. II. definite integrals Def. of a Definite Integral: 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 This is called the definite integral. upper limit lower limit

5. *An indefinite integral is a family of functions, as seen in section 4.1. A definite integral is a number value. Thm. 4.4: Continuity Implies Integrability: If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].

6. Ex. 1: Evaluate by the limit definition. *You will need to use the following formulas to evaluate these kinds of limits.

7. Thm. 4.5: 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 =.

8. Ex. 2: Sketch a region that corresponds to each definite integral. Then evaluate the integral using a geometric formula. a. b. c.

9. III. properties of definite integrals Defs. of Two Special Definite Integrals: 1. If f is defined at x = a, then we define. 2. If f is integrable on [a, b], then we define. Thm. 4.6: Additive Interval Property: If f is integrable on the three closed intervals determined by a, b, and c, where a<c<b, then.

10. Thm. 4.7: Properties of Definite Integrals: If f and g are integrable on [a, b] and k is a constant, then the functions of and are integrable on [a, b], and 1. 2. Thm. 4.8: Preservation of Inequality: 1. If f is integrable and nonnegative on the closed interval [a, b], then. 2. If f and g are integrable on the closed interval [a, b] and for every x in [a, b], then.

11. Ex. 3:Evaluate each definite integral using the following values. a. b. c. d.