Chapter 4 notes from the Text Book

Section 4.2  Area

Theorem 4.2 Summation Formulas  (upper bounds on each is n, lower bounds on each is  ί = 1)

1.  Σ c = cn                                                            2. Σ ί = (n*(n + 1))/ 2    

 3. Σ ί ^2 = (n*(n + 1)*(2n+1))/6                            4. Σ ί ^3 = (n^2*(n + 1)^2)/4

Evaluating a sum  ( the upper bounds is n and the lower bounds is ί )

Σ ((ί +1)/n^2)  for n = 10, 100, 1000, and 10000.

Solution:  Applying Theorem 4.2, you can write

Σ ((ί +1)/n^2)  = (1/n^2) Σ (ί +1)     Factor constant 1/n^2 out of the sum

= (1/n^2) (Σ ί + Σ 1 )                         write as two sums

= (1/n^2) [((n*(n + 1))/2) + n ]                 Apply Theorem 4.2

= (1/n^2) [((n^2) + 3n)/2 ]                       Simplify

= (n+3)/2n                                            Simplify

** Now you can evaluate the sum by substitution the appropraite values of n

n = 10 à 0.65000,  n = 100 à 0.51500, n = 1,000 à 0.50150, n = 10,000 à 0.50015

lim( nà ∞) ((n+3)/2n) = 1/2 

Theorem 4.3  Limit of the Lower and Upper Sums

Let f be continous and nonnegative on the interval [a,b].  The limits as ( nà ∞) of both the lower and upper sums exist and are equal to each other.  That is,

lim( nà ∞)  s(n)  =  lim( nà ∞) Σ f(m _ί) ∆x

                             = lim( nà ∞) Σ f(M _ί) ∆x

                              = lim( nà ∞) S(n)

Where ∆x = (b-a)/n and f(m _ί) and f(M _ί) are the minimum and the maximum values of f on the subinterval.

Definition of the Area of a Region in the Plane

Let f be continous and nonnegative on the interval [a,b].  The area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is

Area = lim( nà ∞)  Σ f(c _ί) ∆x.         x _ί-1  £  c _ί  £ x _ί

where ∆x = (b-a)/n