13.7 Cylindrical Coordinates

13.7 Cylindrical Coordinates - 1 3 .7 
C y l in d r ic a...

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Unformatted text preview: 1 3 .7 
C y l in d r ic a l 
C o o r d in a t e s 
 C y l in d r ic a l 
 z H r, q , zL y x 
 r 
≥ 
0 ,

r 
is 
t h e 
d is t a n c e 
fr o m 
t h e 
o r ig in 
t o 
t h e 
p r o j e c t io n 
 P ′ 
in 
t h e 
x y 
p l a n e .
 0 ≤ θ ≤ 2π , θ 
is
the
angle
from
the
x‐axis
to
the
line
 b e t w e e n 
t h e 
o r ig in 
a n d 
p r o je c t io n 
o f
t h e 
p o in t 
( P ′ ) 
in 
 t h e 
x y 
p l a n e .
 
 C o n v e r s i o n 
 e q u a t i o n s :
 x = r cosθ y tan θ = x y = r sin θ z=z x 2 + y 2 = r2 
 E x .

G r a p h 
r 
= 
1 
 
 
 
 
 
 
 E x .

G r a p h 
 θ = π 4
 
 
 
 
 
 
 E x .

G r a p h 
z 
= 
‐ 2 
 
 
 
 
 E x .
C o n v e r t 
t h e 
e q u a t io n 
6 x 
= 
x 2
+ 
y 2
t o 
c y l in d r ic a l 
 c o o r d in a t e s .
 
 
 
 
 
 
 
 
 
 3− z r= E x .

C o n v e r t 
 2 sin θ 
to
Cartesian
coordinates.
 
 
 
 
 
 
 
 
 
 ∫∫∫ f (r,θ, z) E dV = ∫∫∫ f ( r,θ, z) _________ E 
 
 Ex.
Convert
the
following
integral
to
cylindrical
coordinates
 
 1 ∫∫ 1− x 2 −1 − 1− x 
 
 
 
 
 
 
 
 
 
 
 
 2 ∫ 2− x 2 − y 2 x +y 2 2 (x 2 +y 3 22 ) dzdydx 
 Ex.

Suppose
the
solid
looks
like
 
 
 
 
Ex.

Find
the
volume
of
the
solid
bounded
by
x2
+
y2
=
z2
and

 x2
+
y2
=
4.
 
 
 
 
 
 
 
 
 
 
 
 
 
 ∫∫∫ x Ex.

Set
up
integrals
to
evaluate
 E dV 
where
E
is
the
 solid
bounded
by
x2
+
y2
=
z
and
z
=
18
–
x2
–
y2.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Set
up
integrals
to
find
the
center
of
mass
of
the
solid
 bounded
by
x2
+
y2
=
2y,

z
=
y
+
1,
and
z
=
0.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Do:
Find
the
volume
of
the
solid
inside
the
sphere
x2
+
y2
+
z2
=
 2
and
outside
the
cylinder
x2
+
y2
=
1
using
cylindrical
 coordinates.
 
 
 
 
 
 
 
 ...
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