# 4.3 Riemann Sums and Definite Integrals Definition of 4.3 Riemann Sums and Definite Integrals Definition of the Definite Integral If f is defined on the closed interval [a, b] and the limit of a Riemann sum of f exists, then we say f is integrable on [a, b] and we denote the limit by n b lim f (ci )xi =f ( x)dx x 0 i =1 a 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.

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 b area =f ( x)dx a Areas of common geometric figures. 4 f(x) = 4 3 4dx Area = 4(2) = 8 1

1 2 3 2 4 x 2 dx 2 1 2 1 2 A = r = 2 = 2 2 2 -2 2 Definition of Two Special Definite Integrals

1. If f is defined at x = a, then a f ( x)dx =0 a 2. If f is integrable on [a, b], then a b f ( x)dx =f ( x)dx b a Additive Interval Property If f is integrable on the three closed intervals determined by a, b, and c, then b c

b f ( x)dx =f ( x)dx + f ( x)dx a a a c c b Properties of Basic Integrals b b kf ( x)dx =k f ( x)dx

a b a b b [ f ( x) g ( x)]dx =f ( x)dx g ( x)dx a a a