Chapter 4 notes from the Text Book Section 4.3 Riemann Sums and Definite Integrals Definition of Riemann Sum  Let f be defined on the closed interval [a,b], and let Δ be a partition of [a,b] given by a = x0 < x1 < x2 < x3 < . . . . < xn-1 < xn = b where Δxί  is the width of the ίth subinterval.  If C ί  is any point in the ίth subinterval, the then sum  ( when the upper boundary is n and the lower boundary is ί=1) Σ f(C ί  ) Δxί,           xn-1  ≤ C ί   ≤ xί,        is called a Riemann sum of f for the partition Δ. || Δ || = Δx = (b-a) / n                  regular partition Definition of a Definite Integral If f  is defined on the closed interval [a,b] and the limit lim(|| Δ || à 0) Σ f(C ί  ) Δxί    =  ∫ a b f(x) dx   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. Theorem 4.4     Continuity Implies Integrability If a function f is continuous on the closed interval [a,b], then f is integrable on the [a,b]. Theorem 4.5     The Definite Integral as the Area of a Region If  f is continuous and nonnegative on the close interval [a,b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a               Area =  ∫ a b f(x) dx Example 6  Evaluation of a Definite Integral   Evaluate ∫ 1 3 (-x^2 + 4x -3) dx   using each of the fallowing values. ∫ 1 3 (x^2) dx = 26/3,    ∫ 1 3 (x) dx =  4,     ∫ 1 3  dx = 2   Solution ∫ 1 3 (-x^2 + 4x -3) dx   = ∫ 1 3 (-x^2) dx   + ∫ 1 3 (4x ) dx + ∫ 1 3 (-3) dx                                         = -∫ 1 3 (-x^2) dx   + 4∫ 1 3  dx  - 3 ∫ 1 3  dx                                     = -(26/3) + 4(4)  -3(2)   = (4/3)