Definite Integrals

    In order to understand Riemann sums and integration theory correctly it is important to understand summations using sigma notation.

Definition (Sigma Notation) If are real numbers such that then the summation of these numbers written in sigma notation is



and also using functional notation,



where The is called the index of summation, the is called the th term of the sum, and the upper and lower bounds of the summation are and respectively.

Example (Sigma Notation) Write the sum in expanded form.

    Solution. In expanded form, the sum is

Example (Sigma Notation) Write the sum in expanded form.

    Solution. In expanded form, the sum is

Example (Sigma Notation) Write the sum   in sigma notation.

    Solution. We find that
    

Example (Sigma Notation) Write the sum in sigma notation.

    Solution.  In sigma notation, we have

Example (Sigma Notation) Write the sum



in sigma notation.

    Solution. We find that
    

    

Example (Sigma Notation) Write the sum



in sigma notation.

    Solution. We find that

Proposition (Basic Rules for Sums) If and are real numbers that do not depend on integers and , then

(i) Constant Term Rule:



(ii) Sum Rule:



(iii)  Scalar Multiple Rule:



(iv)  Linearity Rule:



(v) Subtotal Rule: If then



(vi) Dominance Rule: If for all then

Proposition (Summation Formulas) The summation formulas for when















More summations are listed below.

Example (Summation Formulas) Find the value of the sum

    Solution. We find that

Example (Summation Formulas) Find the value of the sum

    Solution. We find that

Example (Summation Formulas) Find the value of the sum

    Solution. We find that

    It is important to be able to evaluate limits of sums so here are a couple of examples.

Example (Summation Formulas) Evaluate the limit  



    Solution. We find that

Example (Summation Formulas) Evaluate the limit



    Solution. We find that
    

Definition (Riemann Sum) A Riemann sum for a function on the closed bounded interval is a sum of the form



where



and for,

    and     

The set is called a partition of and the largest of the is called the norm of .

Example (Riemann Sum) Given the function the closed bounded interval and the partition compute a Riemann sum.

    Solution. Organizing into a table we compute the values,  



So the Riemann sum for these is




  



  

Example (Riemann Sum and Area) Use a Riemann sum to approximate the area under the graph of on with 8 subintervals.

    Solution. As a partition we choose, Organizing our computations and choices for our subinterval representations we have,   
    


So the Riemann sum for this partition and these is




  

Example (Riemann Sum and Area) Use a Riemann sum to approximate the area under the graph of on with 10 subintervals.

    Solution. As a partition we choose,
    

    
Organizing our computations and choices for our subinterval representations we have,   
    




  

    In this topic we illustrate how a Riemann sum can be used to approximate the area under a curve and in doing so, we anticipate the notion of definite integral. We will investigate the area under the curve above the -axis and between the vertical lines and Here is a sketch with the region shaded:



Next let's recall the definition of a Riemann sum.

Definition (Riemann Sum) A Riemann sum for a function on the closed bounded interval is a sum of the form



where



and for,

    and     

The set is called a partition of and the largest of the is called the norm of and the are called the subinterval representatives.

    For our first example we will use a partition with low cardinality, say a partition from using only 4 subintervals between and Also, we will use left-endpoints for our subinterval representatives. Our initial estimate for the area is 20 as shown:

Example (Riemann Sums and Area) Given the function the closed bounded interval and the partition compute a Riemann sum to approximate the area.

    Solution. Organizing into a table we compute the values,  



So the Riemann sum for these is


  




The following sketch shows the Riemann sum as the approximate area under the given curve.



  

    For our second example we will use a finer partition, say a partition using 14 subintervals between and (also with uniform width). We will still use left-endpoints for our subinterval representatives. Our second estimate for the area is as shown:

Example (Riemann Sums and Area) Given the function the closed bounded interval and the partition



compute a Riemann sum to approximate the area.

    Solution. Organizing into a table we compute the values,  



So the Riemann sum for these is


  




The following sketch shows the Riemann sum as the approximate area under the given curve.


  

    Here is an animation that demonstrates what's going on:

    
    

Proposition (Area as the Limit of a Sum) Suppose is continuous and throughout the interval Then the area of the region under the curve over this interval is



where

Example (Area as the Limit of a Sum) Find the exact area under the curve on

    Solution. We will use the formula
    


with and We see and we notice that as and so the area is given by
    
    













(after algebraic simplification)



Therefore, the exact area under the curve bounded by the lines and is Here is a sketch of the region with the area shaded:

Example (Area as the Limit of a Sum) Find the exact area under the curve on   

    Solution. We will use the formula
    


with and We see and we notice that as and so the area is given by
    
    









(after algebraic simplification)



Therefore, the exact area under the curve bounded by the lines and is Here is a sketch of the region with the area shaded:

    Recall a Riemann sum for a function on the closed bounded interval is a sum of the form



where and for,

    and     

The set is called a partition of and the largest of the is called the norm of and is denoted by

Definition (Definite Integral) If is defined on the closed interval and if



exists, then this limits is called the definite integral of from to The definite integral is denoted by

    The function that is being integrated is called the integrand; the interval is the interval of integration; and the endpoints and are called, respectively the lower and upper limits of integration.  

Proposition (Definite Integral of a Continuous Function) If is a continuous function on an interval then is integrable on

Example (Evaluating a Definite Integral using the Definition) Evaluate



using the definition of the definite integral.

    Solution. We will use a formula based on equal width subintervals and right endpoints,  
    


with and We see and we notice that as and so the definte integral is given by
    
    











(after algebraic simplification)

Proposition (Properties of the Definite Integral) Suppose and are integrable on

    (i) (Linearity Rule) The function is integrable on and
    
    
    
    (ii) (Dominance Rule) If throughout then



    (iii) (Subdivision Rule) For any number beteen and
    

Example (Properties of the Definite Integral) If and what is

    Solution. Since is integrable on we can use the Subdivision Rule,



which means

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Cite this as:
Definite Integrals
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/definite-integrals.html