Math 2224 Multivariable Calculus–Sec.13.7: Triple Integrals in Cylindrical and Spherical
Coordinates
I.
Integration in Cylindrical Coordinates
Cylindrical Coordinates are obtained by combining polar coordinates in the
xy
plane with the
usual zaxis.
A. Definition
Cylindrical coordinates represent a point
P
in space by ordered triples
(
r,
θ
,z
)
in which
1.
r
and
θ
are polar coordinates for the vertical projection of
P
on the
xy
plane
2.
z
is the rectangular vertical coordinate.
B. Equations Relating Rectangular and Cylindrical Coordinates
x
=
r
cos
!
,
y
=
r
sin
!
,
z
=
z
x
2
+
y
2
=
r
2
,
tan
!
=
y
x
C. Descriptions of
r=a,
θ
=
θ
0
, z=z
0
1.
r=a
describes a cylinder of radius
a
about the
z
–axis
2.
θ
=
θ
0
describes a plane that contains the
z
–axis and
makes an angle
θ
0
with the positive
x
axis.
3.
z=z
0
describes a plane perpendicular to the
z
–axis
D.
Integrals in Cylindrical Form
1.
Volume of a wedge:
!
V
=
!
z
r
!
r
!
"
2. Volume:
V
=
dV
D
!!!
=
dz rdr d
"
D
!!!
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 Spring '03
 MECothren
 Multivariable Calculus, Polar Coordinates, Polar coordinate system, cylindrical coordinates, cylindrical form

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