Basics of Double Integrals (continued)

 

 

A Type 2 region occurs when integration with respect to the x-coordinate comes first

followed by integration in the y-direction.  The figures below show Type 2 regions.  You

can think of  the process as “sweeping out” the area using the element of area, dA.  Here

you  “sweep” the entire domain, D, first in the x-direction followed by the y-direction.

 

 

                                     

           

                                       

 

 

Strategy recommended for evaluation of double integrals is as follows.

 

a.       Identify the type of Region.  Recall:

b.      Integrate first in the y-direction for Type 1 Regions and in the x-direction for

Type 2 Regions.

c.       Determine the limits of integration then integrate. (Type 1, Type 2 shown below.)

                              Type 1                                     Type 2

                      x=b       y= g2(x)                        y=d      x =  h2(y)

d.      Evaluate  A =  ∫             ∫    dy dx              A =  ∫             ∫    dx dy

                      x=a      y = g1(x)                        y=b       x = h1(y)

 

 

Click for examples.

 



Copyright © 2015 Richard C. Coddington

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