REPEATED INTEGRALS : #repeated #integrals

 Repeated Integral  :

                                 Let f(x) be a bounded function defined on the rectangle R=[a,b;c,d]. 

          Suppose  ∫ f(x ,y) dy exists for each x Ꜫ [a ,b] and y limits from c to d. If g(x) = ∫ f( x ,y )dy, y limits from c to d , then g(x) is a function on [a ,b] . If g(x) is also integrable on [a ,b] , then ∫g(x)dx = ∫[ ∫ f(x , y)dy ] dx is called a repeated integral of f(x ,y) on R where x limits from a to b and y limits from c to d.

                                                          

Problems on Repeated Integrals :

1. Evaluate ∫[ ∫ (x+y+1) dx ] dy and ∫[∫(x+y+1) dy]dx , x limits from -1 to 1 and y limits are from -1 to 0.

2. Show that for the function f(x ,y) defined on [ 0,1;0,1]  with f( x ,y ) = 1/2, y is rational and f( x , y )= x, y  is                  irrational, one of the repeated integral exists and the other does not exist.

3. Examine the nature of the repeated integral  for the function f( x , y ) = 1/y^2 , if 0<x<y<1 and 

            f( x , y )= -1/x^2 , if 0<y<x<1

SOLUTIONS :

1.Evaluate ∫[ ∫ (x+y+1) dx ] dy and ∫[∫(x+y+1) dy]dx , x limits from -1 to 1 and y limits are       from -1 to 0.

Sol :

         

 

2. Show that for the function f(x ,y) defined on [ 0,1;0,1]  with f( x ,y ) = 1/2, y is rational          and f( x , y )= x, y  is irrational, one of the repeated integral exists and the other does not        exist.

Sol :

       

   3.  Examine the nature of the repeated integral  for the function f( x , y ) = 1/y^2 , if 0<x<y<1 and f( x , y )= -1/x^2 , if 0<y<x<1

Sol :

            

                                                           

                                                         

 

 

#repeated #integrals