Fundamental Theorem of Calculus

 

Motivation:

                       Taking the limit of Riemann sum directly is not a easy process although people can use computer to perform this computation.  It is natural that people look for solutions to represent the integral as a function and check if there is any nice property on this kind of function.

 

 

 

 

 

 

Theorem ( Mean Value Theorem for Integral )

          Let  f  be a continuous function on [a,b].   Then there exists c in [a,b] such that

 

                                       f(c)(b-a) =

  Proof:

                     f  is a continuous function on [a, b]  . Thus,  The maximum and minimum exist in [a,b] , denoted as M and m respectively.   So,

                                    m(b-a)    M(b-a) 

                             and

                                        m   f(x)  M  for x in [a,b]

 

                   Again,  f(x) is continuous on a closed interval [a,b] .  So, there exists c in [a,b] such that

                                       f(c) =d           for any d satisfying   m  d  M  .

 

                                          m     M

 

                   The value is between m and M.  Hence there exists c in [a,b] that

                                             f(c) =  

                     i.e.,              f(c)(b-a) =

 

 

 

 

 

Theorem ( Fundamental Theorem of Calculus )

            Let  f  be a continuous function on [a,b] .  For any x in [a,b], define

 

                                  F(x) =

             Then 

(1)      F’(x) =  f(x)  .

(2)    F(d) – F(c) =     for any c, d in [a,b] .

 

 

   Proof:

               (1)

                                    F(x+h) – F(x) =  -

                                                          =

 

                 Since  f  is a continuous function on [a,b], there exists  c in [x,x+h] such that

 

                                                   f(c)h =

                 i.e.,              F(x+h) – F(x) =  f(c)h     for c in [x, x+h]  .

                                     = f(c)   

 

                   When  h ,  f(c)  f(x) .  Hence,

 

                                       F’(x) =  = f(x)

 

               (2)

                             F(c) =  ,  and  F(d) =

 

                        If  d > c,  then  it is natural to have

                                       =  +

                                    -  =

                                  F(d) – F(c) =

 

                        If  c > d,  then

                                     =  +

                                 -  =

                                     F(c) – F(d) =

 

                      However,

                                           +  =  = 0

                                = -

 

                      Thus, we still have

                                       F(d) – F(c) =

 

 

 

                  The Fundamental Theorem of Calculus just bridges the world of  Derivative functions and Integral .  To find out the integral of a function in a closed interval,  it can be done via finding the function with derivative that is equal to the function that we want to perform integration.  It is known as “anti-derivative” function. However,  sometimes it is not easy to find a function to satisfy this relationship.  In that case, we would count on using Riemann sum by computer to figure out the approximate answer.

 

              Furthermore, when we have  F’(x) = f(x) in the theorem above,  we want to represent the result as

                                    F(x) =  + C

                                        where C is just a constant that is independent of x .

 

              This is known as the indefinite form of integral.   It shows that F(x) and the integral  on differs in constant term .  With the relationship shown, once we have the interval for the integral,  we can quickly get the result from F(x).

 

 

 

Example:   

         Sol:

                          cos(t) = -sin(t)

                So,       = (-cos(  )) –(- cos(0)) =  1-(-1) = 2

 

 

 

 

 

Example:  

         Sol:

                       Let  G(x)=  x³ .   Then   G(x) = x²  .

                     = G(1) – G(0) =

 

 

 

 

 

Example:     

         Sol:

                        It is difficult to solve this problem immediately after learning Fundamental Theorem of Calculus.   We want to find out the function with derivative equal to  .   If you can memorize

                                        arcsin(x) =  ,

                  that means you study very hard on the derivative of inverse trigonometric functions.  However, memorizing an approach is better than just memorizing a formula. Of course, there is nothing wrong to memorize formula when you are preparing for an exam that you have to finish a lot of questions in the limited time frame in an exam.  We will introduce how to analyze this kind of problems via change of variable.

 

                            = arcsin(1) – arcsin(0) =  /2