13.7 Spherical Coordinates

13.7 Spherical Coordinates - 1 3 .7 
S p h e r ic a l...

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Unformatted text preview: 1 3 .7 
S p h e r ic a l 
C o o r d in a t e s 
 S p h e r ic a l 
 z y x 
 ρ 
≥ 
0 ,
 ρ 
is 
t h e 
d is t a n c e 
o f
t h e 
p o in t 
t o 
t h e 
o r ig in .
 θ 
is 
t h e 
p o s it iv e 
a n g l e 
b e t w e e n 
t h e 
p o s it iv e 
x 
a x is 
a n d 
 the
line
segment
 OP ′ .
 0 ≤ φ ≤ π ,
 φ 
is 
t h e 
p o s it iv e 
a n g l e 
b e t w e e n 
t h e 
 p o s it iv e 
z 
a x is 
a n d 
t h e 
l in e 
s e g m e n t 
O P .
 
 C o n v e r s i o n 
 e q u a t i o n s :
 z = ρ cosφ r = ρ sin φ x 2 + y 2 + z 2 = ρ2 x = ρ sin φ cos θ y = ρ sin φ sin θ x 2 + y 2 = ρ 2 sin 2 φ 
 Ex.

Graph
 ρ = 3.
 
 
 
 3π E x .

G r a p h 
 θ = 4 .
 
 
 
 E x .

G r a p h 
 φ = 3π 4 .
 
 
 
 
 
 E x .

G r a p h 
 φ = π .
 
 
 
 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 
s p h e r ic a l 
 c o o r d in a t e s .
 
 
 
 
 
 
 
 
 
 E x .
x 2
+ 
y 2
+ 
z 2
= 
1 
 
 
 E x .

z 
= 
√ 2 
 
 
 
 
 
 E x .
( x 
– 
1 ) 2
+ 
y 2
+ 
z 2
= 
1 
 
 
 
 
 
 
 
 Ex.

Convert
 ρ = 3
to
Cartesian
coordinates.
 
 
 
 E x .
 φ = 
 
 
 
 
 π 3
 
 
 
 ∫∫∫ E f ( ρ,φ,θ ) dV = ∫∫∫ f ( ρ,φ,θ ) ρ 2 sin φ dρdθdφ E 
 F in d 
 φ 
a n d 
 ρ 
in 
t h e 
y z 
p l a n e 
 φ 
m e a s u r e s 
d o w n 
fr o m 
t h e 
p o s it iv e 
z ‐ a x is .
 ρ 
m e a s u r e s 
o u t 
fr o m 
t h e 
o r ig in 
t h r o u g h 
t h e 
r e g io n .
 θ 
is 
m e a s u r e d 

o n 
t h e 
x y 
p l a n e .
 
 E x .

C h a n g e 
t o 
s p h e r ic a l 
c o o r d in a t e s 
 3 ∫∫ 9− x 2 −3 − 9 − x 2 ∫ 0 9− x 2 − y 2 z x 2 + y 2 + z 2 dzdydx 
 
 
 
 
 
 
 E x .

S e t 
u p 
in t e g r a l s 
t o 
fin d 
t h e 
v o l u m e 
o f
t h e 
s o l id 
 b o u n d e d 
 b y 
 x 2 
 + 
 y 2 
 = 
 z 2 
 a n d 
 x 2 
 + 
 y 2 

 = 
 4 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Evaluate
 ∫∫∫ y E dV 
where
E
is
a
hollow
sphere
whose
 outside
radius
is
√2
and
the
inside
radius
is
1.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Set
up
integrals
to
find
the
volume
of
the
solid
bounded
 above
by
x2
+
y2
+
z2
=
4z
and
below
by
x2
+
y2
=
z2.
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Set
up
integrals
to
find
the
z
coordinate
of
the
center
of
 mass
of
the
solid
in
the
previous
example.
 
 
 
 
 
 
 Ex.

Set
up
integrals
to
find
the
volume
of
the
solid
inside

 x2
+
y2
+
z2
=
2
and

outside
x2
+
y2
=
1.
 
 
 
 
 
 
 
 Ex.

Set
up
integrals
to
find
the
volume
of
the
solid
bounded
 above
by
z
=
x2
+
y2

and
z
=
3.
 
 
 
 
 
 
 
 
 
 
 
 
 
 Do:

1.
Find
the
volume
of
the
sphere
with
radius
a.
 
 2.

Set
up
an
integral
to
find
the
volume
of
just
the
cone
in
 the
example
using
x2
+
y2
+
z2
=
4z
and
below
by
x2
+
y2
=
z2.
 
 3.

Set
up
integrals
to
find
the
volume
under
the
cone
and
 within
the
sphere
in
the
same
example.
 
 2.
 
 
 
 
 
 3.
 
 
 
 
 ...
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