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 || 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)