Riemann sums & definite integrals (4.3) November 10th, 2016

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 .

The norm of the partition is the largest subinterval and is denoted by *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 .

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

An indefinite integral is a family of functions, as seen in section 4 *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].

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

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 = .

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 .

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 .

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