Changing Order of Integration for a Double Integral

 

Example:  Reverse the order of integration for the integral shown below.

 

                         2    ln x

                 I  =  ∫      ∫   f(x,y) dy dx

                        1     0

 

 

 

 

Strategy:

 

Step 1

Draw the figure using the limits of integration.  (See the figure below.)

Step 2

Identify the intersection of the curves first in x and then in y directions.

Step 3

Use these points of intersection to establish the limits of integration

in reverse order.

  

i.e.  "sweep" element of area, dA, first in the x-direction then in the y-direction.

 

 

The limits of integration are   y = 0  to  y = ln x  and  x = 1 to x = 2.

 

                                                        

 

Integration in the x-direction:            ey    ≤   x    ≤   2

 

Integration in the y-direction:            0   ≤   y  ≤   ln 2

 

                 y = ln 2      x = 2

So       I  =   ∫                 ∫  f(x,y) dx dy                              (result)

                y = 0          x = ey

 

 



Copyright © 2017 Richard C. Coddington

All rights reserved.