Calculus 3 Timed Quiz 5

Proposition (Properties of Double Integrals) Assume that all the given integrals __________ on a rectangular region

(i)  Linearity Rule: For constants and
    
________________  
    
(ii) Dominance Rule:  If _______  throughout a rectangular region then
    


(iii) Subdivision Rule:  If the rectangular region of integration is subdivided into two (disjoint) subrectangles and then

A type I, or _________________ simple region is a region of the plane that can be described by the inequalities  



where and are continuous functions of on Similarly, a type II, or ______________________ simple region , in the plane is a region that can be described by the inequalities  



where and are continuous functions of on

Proposition (Double Integral over a Region) If is a type I region, then



whenever both integrals exist. Similarly, for a type II region



whenever both integrals exist.

Proposition (Double Integral in Polar Coordinates) If is continuous in the polar region described by and (with ), then



Thus the procedure for changing from a Cartesian integral into a polar integral requires substitution of



into the Cartesian integral and then converting the region of integration to polar form  

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