15.3 Double Integrals over General Region

Evaluating Double Integrals over General (Non-Rectangular) Regions

We will consider two types of regions

1

Type I region

A region bounded by

the lines ,x a x b= = and

the curves 1 2( ), ( )y g x y g x= = with [ ]1 2( ) ( ) ,g x g x x a b

Type I region is

1 2{( , ) : , ( ) ( )VR x y a x b g x y g x=

2 ( )y g x=

1( )y g x= X

Type II region

A region bounded by

the lines ,y c y d= = and

the curves 1 2( ), ( )x h y x h y= = with [ ]1 2( ) ( ) ,h y h y y c d

Type II region is

1 2{( , ) : , ( ) ( )}HR x y c y d h y x h y=

x = a x = b

X

Y

2 ( )x h y=

y c=

y d=

1( )x h y=

R

R

2

If R is type I then

2

1

( )

( )( , ) ( , )

b g x

a g xR

f x y dA f x y dydx= If R is type II then

2

1

( )

( )( , ) ( , )

d h y

c h yR

f x y dA f x y dxdy=

Step 2. Move the line left and then right. Leftmost position where the line intersects the

region R is x a= which is lower limit of the outer integral. Rightmost position where the line intersects the region R is x b= which is the upper limit of the outer integral.

Find limit of Integrats for type II region R

Step 1. Draw a horizontal line through the region R at an arbitrary fixed value y. This

line intersects the region left at the curve 1( )y h x= and right at the curve . 2 ( )y h x=Then is the lower limit and 1( )y h x= 1( )y h x= is the upper limit of the inner integral.

Step 2. Move the line down and then up. Lowest position where the line intersects the

region R is which is lower limit of the outer integral. Highest position where the

line intersects the region R is which is the upper limit of the outer integral.

y c=y d=

Step 1. Draw a vertical line through the region R at an arbitrary fixed value x. This line

intersects the region below at the curve 1( )y g x= and above at the curve . 2 ( )y g x=Then is the lower limit and 1( )y g x= 1( )y g x= is the upper limit of the inner integral.

Find limit of Integrals for type I region R

Double integrals over both regions

evaluated as iterated integrals

Important points to note:

Outer limits always constant The idea of using a vertical line as explained in class Sometimes need to break into more integrals.

Question 5/1002: Evaluate /2 cos

sin

0 0

e drd

.

Question 9/1002: Evaluate the double integral 22

1R

y dAx +

where {( , ) : 0 1,0 }R x y x y x= .

Question 17/1002: Evaluate the double integral (2 )R

x y dA ; R is bounded by the circle with center the origin and radius 2.

Question 22/1002: Use double integral to find the volume of the solidenclosed by

the paraboloid 2 3 2z x y= + and the plane 0, 1, , 0x y y x z= = = = .

3

4

2

Question 28/1002: Find the volume of the region bounded by the cylinders

2 2x y r+ = and 2 2 2y z r+ = .

Question 42/1003: Sketch the region of integration and change the order of the

integration 1 /4

0 arctan

( , )f x y dydx

.

Question 48/1003: Evaluate the integral 4

3

8 2

0

x

y

e dxdy by reversing the order of integration.