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Calculus II Notes 16.3 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 16.3 16.3 Double Integrals over General Regions For a function of a single variable, the region we integrate over is always an interval. This is not true for functions of two or more variables. We can extend the double integral on a rectangular region to more general regions in the following way. Let f ( x , y ) be defined on a bounded region D whose boundary is made up of finitely many simple smooth curves. Let R be a rectangular region containing D . Define F ( x , y ) = f ( x , y ) if ( x , y ) D 0 if ( x , y ) R \ D so that f ( x , y ) D ∫∫ dA = F ( x , y ) R ∫∫ dA . Definition The region D R 2 is: Type I if D = ( x , y ) a x b and g 1 ( x ) y g 2 ( x ) { } Type II if D = ( x , y ) c y d and h 1 ( y ) x h 2 ( y ) { } . Theorem If D is type I, then f ( x , y ) D ∫∫ dA = f ( x , y ) g 1 ( x ) g 2 ( x ) a b dy dx , and if D is type II, then f ( x , y ) D ∫∫ dA = f ( x , y ) h 1 ( y ) h 2 ( y ) c d dx dy . Example A For D = ( x , y ) 0 x 1 and 0 y x { } x 3 y D ∫∫ dA = x 3 y 0 x dy ( ) 0 1 dx = x 3 y 2 2 0 1 0 x dx = x 5 2 0 0 1 dx = x 6 12 0 1 = 1 12 . Example B For D = ( x , y ) 0 y 1 and 1 x y { } xy y 3 ( ) D ∫∫ dA = xy y 3 ( ) 1 y dx ( ) 0 1 dy = x 2 y 2 xy 3 0 1
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