# Slide_11 - MA1104 Multivariable Calculus Lecture 11 Dr KU...

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MA1104 Multivariable Calculus Lecture 11 Dr KU Cheng Yeaw Thursday Feb 19, 2009

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Recap Last lecture . .... we deﬁned double integral over a rectangle region R via Fubini’s Theorem, we can evaluate double integral over a rectangle region using iterated integral we extend our deﬁnition of double integral to that that over a more general region D .
Overview In this lecture, we learn how to evaluate double integral over Type I and Type II region we learn how to evaluate double integral over polar rectangle.

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Double Integral over General Region Recall how we have deﬁned double integral of f over a general region D : We ﬁrst deﬁne a new function F ( x,y ) = ± f ( x,y ) if ( x,y ) D 0 if ( x,y ) R - D If F is integrable over R , then we deﬁne the double integral of f over D by ZZ D f ( x,y ) dA = ZZ R F ( x,y ) dA.

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In general, F ( x,y ) may not be integrable. However, we can avoid this problem given some mild conditions. Double Integral Over Bounded Region Let D be a domain whose boundary is a simple, closed, piecewise smooth curve. If f ( x,y ) is continuous on D , then ZZ D f ( x,y ) dA = ZZ R F ( x,y ) dA exists.
Now, we shall focus on two particular types of domain D which is a region between two curves. They satisfy the assumptions in the previous theorem which ensures the integrability of F ( x,y ) . We call these domain to be of Type I and Type II .

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Type I Domain Type I Domain A plane region D is said to be of Type I if it lies between the graphs of two continuous functions of x , that is, D = { ( x,y ) : a x b,g 1 ( x ) y g 2 ( x ) } where g 1 ( x ) and g 2 ( x ) are continuous on [ a,b ] .
Some examples of Type I region:

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How do we compute the integral of f ( x,y ) over such D ? Integral over Type I Domain If f is continuous on a Type I domain D such that D = { ( x,y ) : a x b,g 1 ( x ) y g 2 ( x ) } then ZZ D f ( x,y ) dA = Z b a Z g 2 ( x ) g 1 ( x ) f ( x,y ) dy dx.

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Observe that the expression on the right-hand side of ZZ D f ( x,y ) dA = Z b a Z g 2 ( x ) g 1 ( x ) f ( x,y ) dy dx is an iterated integral similar to the ones we have for rectangle region, except that in the inner integral we regard x as being constant not only in f ( x,y ) but also in the limits of the integration , g 1 ( x ) and g 2 ( x ) .
Lets prove the theorem, well, actually, we are not able to give a complete proof . .. Proof. By deﬁnition, we need to choose a rectangle R = [ a,b ] × [ c,d ] that contains D and set F ( x,y ) = ± f ( x,y ) if ( x,y ) D 0 if ( x,y ) R - D

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Then, ZZ D f ( x,y ) dA = ZZ R F ( x,y ) dA = Z b a Z d c F ( x,y ) dy dx. Remark. Although F ( x,y ) need not be continuous, the use of Fubini’s Theorem in the second equation above can be justiﬁed (however, we will not supply the details here). In particular, R d c F ( x,y ) dy exists and is a continuous function of x .

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Observe that F ( x,y ) = 0 if y < g 1 ( x ) or y > g 2 ( x ) because then ( x,y ) lies outside D . Therefore, Z d c F (
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Slide_11 - MA1104 Multivariable Calculus Lecture 11 Dr KU...

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