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6.3 Course Notes
Triple Integrals in
Cylindrical Coordinates
CYLINDRICAL COORDINATES
RECALL:
In the cylindrical coordinate system, a point P
in three-dimensional (3-D) space is
represented by (r, , z), where:
r and are polar
coordinates of
the projection of P
onto the xyplane.
z is the directed distance
from the xy-plane to P.
TRIPLE INTEGRAL IN CYLINDRICAL COORDINATES
Page 97 of Notes
Write x = r cos , y = r sin , z = t.
Replace the volume element
dx dy dz by r dr d dz
ie.
. and , for limits e appropriat with
) , sin , cos ( ) , , (
t r
dt d dr r t r r dz dy dx z y x
u
u u u | |
}}} }}}
=
dt d dr r dV u =
EVALUATING TRIPLE INTEGRALS
Calculate the mass of the solid lying within:
The cylinder x
2
+ y
2
= 1
Below the plane z = 4
Above the paraboloid
z = 1 x
2
y
2
The density at (x, y, z) is given by:
where K = constant.
Example
( )
2 2
, , f x y z K x y Kr = + =
EVALUATING TRIPLE INTEGRALS
In cylindrical coordinates, the cylinder is
r = 1
and the paraboloid is
z = 1 r
2
.
So, we can write:
V = {(r, , z)| 0 2,
0 r 1,
1 r
2
z 4}
Example EVALUATING TRIPLE INTEGRALS
The mass of V is:
( )
( )
2
2 2
2 1 4
0 0 1
2 1
2 2
0 0
2 1
2 4
0 0
1
5
3
0
( )
4 1
3
12
2
5 5
E
r
m K x y dV
Kr r dz dr d
Kr r dr d
K d r r dr
r K
K r
t
t
t
u
u
u
t
t
= +
=
= (
= +
(
= + =
(
}}}
} } }
} }
} }
Example
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6.4 Course Notes
Triple Integrals
in Spherical Coordinates
SPHERICAL COORDINATES
Recall the spherical coordinates (, , )
of a point P in space:
= |OP| is the distance from the origin to P.
is the same angle as
in cylindrical coordinates.
is the angle between
the positive z-axis and
the line segment OP.
TRIPLE INTEGRALS IN SPHERICAL COORDINATES
The volume element is:
. and , for limits e appropriat with
sin ) cos , sin sin , sin cos (
) , , (
2
t r
d d dr r r r r
dz dy dx z y x
u
0 u 0 0 0 u 0 u |
|
}}}
}}}
=
Page 99
0 u 0 d d dr r dxdydz sin
2
=
Use spherical coordinates to find the
volume of the solid that lies:
Above the cone
Below the sphere
x
2
+ y
2
+ z
2
= z
2 2
z x y = +
Example SPHERICAL COORDINATES
Notice that the sphere passes through
the origin and has center (0, 0, ).
We write its equation
in spherical
coordinates as:
2
= cos
or
= cos
Example SPHERICAL COORDINATES
The equation of the cone can be written
as:
This gives:
sin= cos
or
= /4
2 2 2 2 2 2
cos sin cos sin sin
sin
| | u | u
|
= +
=
Example SPHERICAL COORDINATES
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Thus, the description of the solid E
in spherical coordinates is:
( ) { }
, , 0 2 , 0 / 4, 0 cos
E
u | u t | t |
=
s s s s s s
Example SPHERICAL COORDINATES
The volume of E is:
2 / 4 cos
2
0 0 0
cos
3
2 / 4
0 0
0
/ 4
3
0
/ 4
4
0
( ) sin
sin
3
2
sin cos
3
2 cos
3 4 8
E
V E dV d d d
d d
d
t t |
|
t t
t
t
| | u
u | |
t
| | |
t | t
=
=
= =
(
=
(
=
(
= =
(
}}} } } }
} }
}
SPHERICAL COORDINATES Example
More Examples on
Triple Integrals
EVALUATING TRIPLE INTEGRALS
Evaluate
This is a triple integral over the solid region
Example
( )
2
2 2 2
2 4 2
2 2
2 4
x
x x y
x y dz dy dx
+
+
} } }
( )
2 2 2 2
{ , , | 2 2, 4 4 , 2}
E
x y z x x y x x y z
=
s s s s + s s
EVALUATING TRIPLE INTEGRALS
The projection of E onto the xy-plane
is the disk x
2
+ y
2
4.
The lower surface of
E is the cone
Its upper surface is
the plane z = 2.
Example
2 2
z x y = +
EVALUATING TRIPLE INTEGRALS
That region has a much simpler description
in cylindrical coordinates:
E =
{(r, , z) | 0 2, 0 r 2, r z 2}
Thus, we have the following result.
Example
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EVALUATING TRIPLE INTEGRALS
( )
( )
( )
2
2 2 2
2 4 2
2 2
2 4
2 2 2
2 2 2
0 0
2 2
3
0 0
2
4 5
1 1
2 5
0
16
5
2
2
x
x x y
r
E
x y dz dy dx
x y dV r r dz dr d
d r r dr
r r
t
t
u
u
t
t
+
+
= + =
=
= (
=
} } }
}}} } } }
} }
Example
Evaluate
where B is the unit sphere:
( )
3/ 2
2 2 2
x y z
B
e dV
+ +
}}}
( )
{ }
2 2 2
, , 1 B x y z x y z = + + s
TRIPLE INTGN. IN SPH. COORDS. Example
As the boundary of B is a sphere, we use
spherical coordinates:
In addition, spherical coordinates are appropriate
because:
x
2
+ y
2
+ z
2
=
2
( ) { }
, , 0 1, 0 2 , 0 B u | u t | t = s s s s s s
Example TRIPLE INTGN. IN SPH. COORDS.
( )
( )
| | ( ) ( )
3/ 2
2 2 2
3/ 2
2
3
3
2 1
2
0 0 0
2 1
2
0 0 0
1
1 4
3 3
0
0
sin
sin
cos 2 1
x y z
B
e dV
e d d d
d d e d
e e
t t
t t
t
| u |
| | u
| t t
+ +
=
=
(
= =
}}}
} } }
} } }
Example TRIPLE INTGN. IN SPH. COORDS.
It would have been extremely awkward to
evaluate the integral without spherical
coordinates.
In rectangular coordinates, the iterated integral
would have been:
( )
3/ 2
2 2 2
2 2 2
2 2 2
1 1 1
1 1 1
x x y
x y z
x x y
e dz dy dx
+ +
} } }
Note TRIPLE INTGN. IN SPH. COORDS.
