5.2  The Definite Integral

 

Definition of Definite Integral  -  Definition 2

          If f is continuous on [a,b].  Divide [a,b] into n subintervals of equal width, .  Let , be the endpoints of the subintervals, and choose sample points,  in the subintervals .  The definite integral of f from a to b is

                    

 

 

Terminology

 

         Integral sign

 

          Integrand

 

          Limits of integration - upper limit, lower limit

 

          Variable of integration      

 

 

A definite integral is a number, the variable "doesn't matter"   

 

 

 

 

As a limit of a continuous function, the particular choice of  doesn't change the results, often use the right or left endpoint, or the midpoint of the interval.

 

 

Riemann Sum  is  

 

 

Area Interpretation

          [image]                                                [image]

 

          If  is always > 0  ….                If is always < 0  ….

 

          [image]

 

          If is both positive and negative on an interval

 

 

Note - we used equal subintervals , but can be interpreted as different sizes and still same results then, 

 

Example

 

Transparency 

 

 

 

 

 

 

 

 

Theorem 3

If f is continuous on [a,b], or if f  has only a finite number of jump discontinuities, then f  is integrable on [a,b]; that is, the definite integral

 exists.

Theorem 4

If f is integrable on [a,b], then

Where 

 

 

Express the limit as a definite integral on the given interval

                           

 

 

 

 

 

 

#12  Uses the midpoint rule with the given value of n to approximate the integral.  Round the answer to four decimal places.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Demonstrate, using a program on calculator to compute for larger values of n.

 

Let’s abstract the process and use limits for a few definite integral.   Necessary formulas:

 

Common Summations

                 

 

 

And Summation Rules 

              

 

Use the form of the definition of the integral given in Theorem 4to evaluate the integral.

 

#22 

 

 

 

 

 

 

 

 

 

 

 

 

 

#24 

 

 

 

 

 

 

 

 

 

 

 

 

 

*an easier method for evaluating definite integrals will be covered in 5.3!

 

Properties of Definite Integrals

                     

 

                      

 

 

       

#34   Transparency

 

 

 

 

 

 

 

PROPERTIES OF THE INTEGRAL

1. 

 

2. 

 

 

3. 

 

4. 

 

5. 

 

Comparison Properies of the Integral  - page 309

 

Evaluate the integral by interpreting it in terms of areas.

#36                                        #38 

 

 

 

 

 

 

#40                                           #50 

 

 

 

 

 

 

#44  Use the properties of integrals and the results of Example 3  ()to evaluate

 

 

 

 

#48  If

 

         

 

 

5.2 pg. 310; 3, 5, 9, 11, 17, 19, 20, 21-25 odd, 29, 33, 35-41 odd, 42, 43, 47, 49