mat 1236 calculus iii section 15.4 double integrals in polar coordinates
TRANSCRIPT
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