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  • Recognize when a function of two variables is integrable over a general region.
  • Evaluate a double integral by computing an iterated integral over a region bounded by two vertical lines and two functions of x , or two horizontal lines and two functions of y .
  • Simplify the calculation of an iterated integral by changing the order of integration.
  • Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region.
  • Solve problems involving double improper integrals.

In Double Integrals over Rectangular Regions , we studied the concept of double integrals and examined the tools needed to compute them. We learned techniques and properties to integrate functions of two variables over rectangular regions. We also discussed several applications, such as finding the volume bounded above by a function over a rectangular region, finding area by integration, and calculating the average value of a function of two variables.

In this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need to be extended to cover this more general case.

General regions of integration

An example of a general bounded region D on a plane is shown in [link] . Since D is bounded on the plane, there must exist a rectangular region R on the same plane that encloses the region D , that is, a rectangular region R exists such that D is a subset of R ( D R ) .

A rectangle R with a shape D inside of it. Inside D, there is a point labeled g(x, y) = f(x, y). Outside D but still inside R, there is a point labeled g(x, y) = 0.
For a region D that is a subset of R , we can define a function g ( x , y ) to equal f ( x , y ) at every point in D and 0 at every point of R not in D .

Suppose z = f ( x , y ) is defined on a general planar bounded region D as in [link] . In order to develop double integrals of f over D , we extend the definition of the function to include all points on the rectangular region R and then use the concepts and tools from the preceding section. But how do we extend the definition of f to include all the points on R ? We do this by defining a new function g ( x , y ) on R as follows:

g ( x , y ) = { f ( x , y ) if ( x , y ) is in D 0 if ( x , y ) is in R but not in D

Note that we might have some technical difficulties if the boundary of D is complicated. So we assume the boundary to be a piecewise smooth and continuous simple closed curve. Also, since all the results developed in Double Integrals over Rectangular Regions used an integrable function f ( x , y ) , we must be careful about g ( x , y ) and verify that g ( x , y ) is an integrable function over the rectangular region R . This happens as long as the region D is bounded by simple closed curves. For now we will concentrate on the descriptions of the regions rather than the function and extend our theory appropriately for integration.

We consider two types of planar bounded regions.

Definition

A region D in the ( x , y ) -plane is of Type I    if it lies between two vertical lines and the graphs of two continuous functions g 1 ( x ) and g 2 ( x ) . That is ( [link] ),

D = { ( x , y ) | a x b , g 1 ( x ) y g 2 ( x ) } .

A region D in the x y plane is of Type II    if it lies between two horizontal lines and the graphs of two continuous functions h 1 ( y ) and h 2 ( y ) . That is ( [link] ),

D = { ( x , y ) | c y d , h 1 ( y ) x h 2 ( y ) } .

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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