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Calculus volume 3 Multiple integration Triple integrals in cylindrical
Key concepts
To evaluate a triple integral in cylindrical coordinates, use the iterated integral
∫
θ
=
α
θ
=
β
∫
r
=
g
1
(
θ
)
r
=
g
2
(
θ
)
∫
z
=
u
1
(
r
,
θ
)
z
=
u
2
(
r
,
θ
)
f
(
r
,
θ
,
z
)
r
d
z
d
r
d
θ
.
To evaluate a triple integral in spherical coordinates, use the iterated integral
∫
θ
=
α
θ
=
β
∫
ρ
=
g
1
(
θ
)
ρ
=
g
2
(
θ
)
∫
φ
=
u
1
(
r
,
θ
)
φ
=
u
2
(
r
,
θ
)
f
(
ρ
,
θ
,
φ
)
ρ
2
sin
φ
d
φ
d
ρ
d
θ
.
Key equations
Triple integral in cylindrical coordinates
∭
B
g
(
x
,
y
,
z
)
d
V
=
∭
B
g
(
r
cos
θ
,
r
sin
θ
,
z
)
r
d
r
d
θ
d
z
=
∭
B
f
(
r
,
θ
,
z
)
r
d
r
d
θ
d
z
Triple integral in spherical coordinates
∭
B
f
(
ρ
,
θ
,
φ
)
ρ
2
sin
φ
d
ρ
d
φ
d
θ
=
∫
φ
=
γ
φ
=
ψ
∫
θ
=
α
θ
=
β
∫
ρ
=
a
ρ
=
b
f
(
ρ
,
θ
,
φ
)
ρ
2
sin
φ
d
ρ
d
φ
d
θ
In the following exercises, evaluate the triple integrals
∭
E
f
(
x
,
y
,
z
)
d
V over the solid
E
.
Let
B be a cylindrical shell with inner radius
a
, outer radius
b
, and height
c
, where
0
<
a
<
b and
c
>
0
. Assume that a function
F defined on
B can be expressed in cylindrical coordinates as
F
(
x
,
y
,
z
)
=
f
(
r
)
+
h
(
z
)
, where
f and
h are differentiable functions. If
∫
a
b
f
˜
(
r
)
d
r
=
0 and
h
˜
(
0
)
=
0
, where
f
˜ and
h
˜ are antiderivatives of
f and
h
, respectively, show that
∭
B
F
(
x
,
y
,
z
)
d
V
=
2
π
c
(
b
f
˜
(
b
)
−
a
f
˜
(
a
)
)
+
π
(
b
2
−
a
2
)
h
˜
(
c
)
.
Use the previous result to show that
∭
B
(
z
+
sin
x
2
+
y
2
)
d
x
d
y
d
z
=
6
π
2
(
π
−
2
)
, where
B is a cylindrical shell with inner radius
π
, outer radius
2
π
, and height
2
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Let
B be a cylindrical shell with inner radius
a
, outer radius
b
, and height
c
, where
0
<
a
<
b and
c
>
0
. Assume that a function
F defined on
B can be expressed in cylindrical coordinates as
F
(
x
,
y
,
z
)
=
f
(
r
)
g
(
θ
)
h
(
z
)
, where
f
,
g
,
and
h are differentiable functions. If
∫
a
b
f
˜
(
r
)
d
r
=
0
, where
f
˜ is an antiderivative of
f
, show that
∭
B
F
(
x
,
y
,
z
)
d
V
=
[
b
f
˜
(
b
)
−
a
f
˜
(
a
)
]
[
g
˜
(
2
π
)
−
g
˜
(
0
)
]
[
h
˜
(
c
)
−
h
˜
(
0
)
]
,
g
˜ and
h
˜ are antiderivatives of
g and
h
, respectively.
Use the previous result to show that
∭
B
z
sin
x
2
+
y
2
d
x
d
y
d
z
=
−12
π
2
, where
B is a cylindrical shell with inner radius
π
, outer radius
2
π
, and height
2
. Got questions? Get instant answers now!
In the following exercises, the boundaries of the solid
E are given in cylindrical coordinates.
Express the region
E in cylindrical coordinates.
Convert the integral
∭
E
f
(
x
,
y
,
z
)
d
V to cylindrical coordinates.
E is bounded by the right circular cylinder
r
=
4
sin
θ
, the
r
θ -plane, and the sphere
r
2
+
z
2
=
16
.
a.
E
=
{
(
r
,
θ
,
z
)
|
0
≤
θ
≤
π
,
0
≤
r
≤
4
sin
θ
,
0
≤
z
≤
16
−
r
2
}
; b.
∫
0
π
∫
0
4
sin
θ
∫
0
16
−
r
2
f
(
r
,
θ
,
z
)
r
d
z
d
r
d
θ
Got questions? Get instant answers now!
E is located in the first octant and is bounded by the circular paraboloid
z
=
9
−
3
r
2
, the cylinder
r
=
3
, and the plane
r
(
cos
θ
+
sin
θ
)
=
20
−
z
.
a.
E
=
{
(
r
,
θ
,
z
)
|
0
≤
θ
≤
π
2
,
0
≤
r
≤
3
,
9
−
r
2
≤
z
≤
10
−
r
(
cos
θ
+
sin
θ
)
}
; b.
∫
0
π
/
2
∫
0
3
∫
9
−
r
2
10
−
r
(
cos
θ
+
sin
θ
)
f
(
r
,
θ
,
z
)
r
d
z
d
r
d
θ
Got questions? Get instant answers now!
E is located in the first octant outside the circular paraboloid
z
=
10
−
2
r
2 and inside the cylinder
r
=
5 and is bounded also by the planes
z
=
20 and
θ
=
π
4
.
Got questions? Get instant answers now!
In the following exercises, the function
f and region
E are given.
Express the region
E and the function
f in cylindrical coordinates.
Convert the integral
∭
B
f
(
x
,
y
,
z
)
d
V into cylindrical coordinates and evaluate it.
f
(
x
,
y
,
z
)
=
1
x
+
3
,
E
=
{
(
x
,
y
,
z
)
|
0
≤
x
2
+
y
2
≤
9
,
x
≥
0
,
y
≥
0
,
0
≤
z
≤
x
+
3
}
a.
E
=
{
(
r
,
θ
,
z
)
|
0
≤
r
≤
3
,
0
≤
θ
≤
π
2
,
0
≤
z
≤
r
cos
θ
+
3
}
,
f
(
r
,
θ
,
z
)
=
1
r
cos
θ
+
3
; b.
∫
0
3
∫
0
π
/
2
∫
0
r
cos
θ
+
3
r
r
cos
θ
+
3
d
z
d
θ
d
r
=
9
π
4
Got questions? Get instant answers now!
Source:
OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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