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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online