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 < . . . . < x_n-1 < x_n = b

where Δx_ί  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  ) Δx_ί,           x_n-1  ≤ C _ί   ≤ x_ί,       

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  ) Δx_ί    =  ∫ _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)