MAT 1236Calculus III

Section 15.4

Double Integrals In Polar Coordinates

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HW & …

WebAssign 15.4 (10 problems, 82 min.)

Preview

Formula for double integral if the region is described in polar coordinates

consider the case where the bounds are all constants (Polar rectangle)

Rectangular and polar regions are in the most popular applications in physics and engineering

Polar Rectangle

} ,|),{( brarR

xO

br

ar

R

Polar Rectangle

{( , ) | 2 3, }6 3

R r r

xO

3r 2r R

6

3

Example A

xR

21

y ( , ) | , R r r

Example B

xR

21

y ( , ) | , R r r

Example C

xR 2

y ( , ) | , R r r

Polar Rectangle

} ,|),{( brarR

xO

br

ar

R( , )z f x y

R

x

y

z

Polar Rectangle

} ,|),{( brarR

xO

br

ar

R

( , ) ?R

f x y dA

( , )z f x y

R

x

y

z

Idea } ,|),{( brarR

xO

R

r

iA

( , )

( cos , sin )i j i j

x y

r r

Idea

xO

br

ar

R( , )z f x y

R

x

y

z

,1 1

( , ) lim ( cos , sin )m n

i j i jm n

iR

ij

f x y d r AA f r

Idea } ,|),{( brarR

xO

R

r

iA

( , )

( cos , sin )i j i j

x y

r r

,1 1

( , ) lim ( cos , sin )m n

i j i jm n

iR

ij

f x y d r AA f r

1

1if

2

i i

i i i

A r r

r r r

Polar Rectangle

xO

br

ar

R

rdrdrrf

dAyxf

b

a

R

)sin,cos(

),(

,1 1

( , ) lim ( cos , sin )m n

i j i j im n

i jR

f x y dA f r r r r

Formula

Order of integration is not important (why?)

rdrdrrfdAyxfb

aR )sin,cos(),(

Splitting Formula

( cos , sin ) ( ) ( ),

( , ) ( cos , sin )

( ) ( )

( ) ( )

b

R a

b

a

b

a

If f r r g r h then

f x y dA f r r rdrd

g r h rdrd

g r dr h dr

Example 1

}2/0 ,21|),{( rrR

R

ydAx2

31

15

Remarks

Sometimes, an integral in polar coordinates may be easier to evaluate than the corresponding one in rectangular coordinates

Example 2

Evaluate

by converting to polar coordinates

1

1

1

0

2/322

2

)(y

dxdyyx

Example 2

2{( , ) | 0 1 , 1 1}R x y x y y

1

1

1

0

2/322

2

)(y

dxdyyx

1

1

1

21 yx 0x

Rx

y

Example 2

1

1

1

Rx

y

{( , ) | 0 1, / 2 / 2}R r r

Example 2

{( , ) | 0 1, / 2 / 2}R r r

2112 2 3/2

1 0

,

( )

( , )

y

R

f x y

x y dxdy

f x y dA

1

1

15

xR

y

Addition Formula for 15.4

1 2{( , ) | , }D r rh h

2

1

( , )

( cos , sin )h

h

D

f x y dA

f r r rdrd