Double Integrals over General Regions Lesson 15.2

University: Santa Monica College

Course: Multivariable Calculus (MATH 11)

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Calc 3 Handout Section 15.2: Double Integrals over General Regions I. Lai

Previously we only looked at double integrals over rectangular regions. Now we want to look at double

integrals over regions that are not rectangular. We will call such a region

D

.

We will consider two types of general regions, which we call Type I and Type II regions.

Type I Type II

***note: all functions are continuous

A type I region is a region that is defined as

( )

û ý

1 2

, | , ( ) ( )D x y a x b g x y g x=óóóó

(region is better sliced vertically)

A type II region is a region that is defined as

( )

û ý

1 2

, | , ( ) ( )D x y c y d h y x h y=óóóó

(region is better sliced horizontally)

If

f

is continuous on a type I region

( )

û ý

1 2

, | , ( ) ( )D x y a x b g x y g x=óóóó

,

then

( ) ( )

2

1

( )

( )

, ,

b g x

a g x

D

f x y dA f x y dy dx=

òò ò ò

If

f

is continuous on a type II region

( )

û ý

1 2

, | , ( ) ( )D x y c y d h y x h y=óóóó

,

then

( ) ( )

2

1

( )

( )

, ,

d h y

c h y

D

f x y dA f x y dx dy=

òò ò ò

***notice that we must first do the integral where the bounds of integration are functions, then do the

integrals where the bounds of integration are constants. This cannot be switched! (Think about why)

***for any region, you can consider it either a type I or type II region, and it would give the same answer.

However, in general, it is easier to look at it as one type over the other, depending on what the region is.

Properties of Double Integrals

1)

( ) ( ) ( ) ( )

, , , ,

D D D

f x y g x y dA f x y dA g x y dA+= +

ùù

ûû

òò òò òò

2)

( ) ( )

, ,

D D

c f x y dA c f x y dA=

òò òò

3) If

1 2

D D D=ø

, that is, the region D is a union of two or more nonoverlapping regions, then

( ) ( ) ( )

1 2

, , ,

D D D

f x y dA f x y dA f x y dA=+

òò òò òò

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dA dydx constants daddy

dA dydx

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