2. 4 .
Cylindrical coordinates
easier way to do triple integrals
C- onto
FIB If projecting
^
r Maish
•
yt plane
-
then cylindrical
< o↳¥ac ,yi◦, ≥
coordinates : of r.ly , 2- )
( r , o, t ) ( x
,
r
, A)
-
↓
•
JE r To > y
rosa
E-
rsino v
1-
Eff Lf projecting f- onto
✗ t plane
-
cyl .
Coard :
,
NZ
( rcoslt, y , rsino) r.cn,z)
↑&
E ↓
e
r ) 2C
✓
*mµµyµµµgµµmµµµ,÷
To write triple integral it -0 .
cylindrical coordinates
◦
Describe projection onto plane i. to .
polar coordinates
•
describe 2- it -0 .
rand 0
•
change function f- In ,y,t) to
gcrcosoyrsino ) ,
t
, Example
cytcoord for
① give iterated integrals in .
the solid bounded by
fffxytdv and C- region
E
z=✓xÑ and 7--3
① ①
curve of intersection
3
M£
⊖ 2- =
✓TÑ3
① ①
y
✗
21yd
-
9
A ↑
↳ci
L
enter one
a from this
perspective
in plane
projection
xy _
→
^ ① Cartesian → ② cylindrical
3
R
0Erk3
←# 3 ≤ * 13
-
1- • ≤ 21T
-
✓ ≤
yytvq.TO
✓
Eye ≤ *≤ 3 → VÑy≤ 2- ≤ 3
③ RT ≤ 2- ≤ s
fffnytdv
✓
r ≤ 2- ≤ 3
>
↳
Crusoe)CrsinMZ]r
if.it/Fj#rcoso)crsinos2-Jrdtdrdo
'
'
'
fir > 2-
?⃝
Cylindrical coordinates
easier way to do triple integrals
C- onto
FIB If projecting
^
r Maish
•
yt plane
-
then cylindrical
< o↳¥ac ,yi◦, ≥
coordinates : of r.ly , 2- )
( r , o, t ) ( x
,
r
, A)
-
↓
•
JE r To > y
rosa
E-
rsino v
1-
Eff Lf projecting f- onto
✗ t plane
-
cyl .
Coard :
,
NZ
( rcoslt, y , rsino) r.cn,z)
↑&
E ↓
e
r ) 2C
✓
*mµµyµµµgµµmµµµ,÷
To write triple integral it -0 .
cylindrical coordinates
◦
Describe projection onto plane i. to .
polar coordinates
•
describe 2- it -0 .
rand 0
•
change function f- In ,y,t) to
gcrcosoyrsino ) ,
t
, Example
cytcoord for
① give iterated integrals in .
the solid bounded by
fffxytdv and C- region
E
z=✓xÑ and 7--3
① ①
curve of intersection
3
M£
⊖ 2- =
✓TÑ3
① ①
y
✗
21yd
-
9
A ↑
↳ci
L
enter one
a from this
perspective
in plane
projection
xy _
→
^ ① Cartesian → ② cylindrical
3
R
0Erk3
←# 3 ≤ * 13
-
1- • ≤ 21T
-
✓ ≤
yytvq.TO
✓
Eye ≤ *≤ 3 → VÑy≤ 2- ≤ 3
③ RT ≤ 2- ≤ s
fffnytdv
✓
r ≤ 2- ≤ 3
>
↳
Crusoe)CrsinMZ]r
if.it/Fj#rcoso)crsinos2-Jrdtdrdo
'
'
'
fir > 2-
?⃝
