Triple Integral in Cylindrical Coordinates
Triple Integrals in Cylindrical coordinates
From Cylindrical to Cartesian:
cossin
x ry rz z
and are the polar coordinates of the projection of the point P onto the -plane.
is the signed vertical distance between P and the -plane (same as in cartesian)
From Cartesian to Cylindrical:2 2 2
tan
r x yyxz z
Cylindrical coordinates of the point P:
Triple Integrals in Cylindrical coordinates
Example 1: Given find x, y and z.
Conversely, given rectangular find cylindrical:
cossin
x ry rz z
,2( , , ) 3, , 13
r z
32
3 32
3cos(2 / 3) 3( 1/ 2)
3sin(2 / 3)
1
x
y
z
1
2 2
(1) 5tan 1, 3rd quadrant tan4
3 27
yx
r x yz
5( , , ) 3 2, ,74
r z is a possible answer
2 2 2
tan
r x yyx
z z
Triple Integrals in Cylindrical Coordinates
Basic graphs in cylindrical coordinates:
represents a cylinder ( in cartesian)
represents a vertical plane (if
r ≥ 0, half a plane)
represents a horizontal plane
represents the cone
Triple Integrals in cylindrical coordinates
Since , integrals involving or frequently are easier in cylindrical coordinates.
The volume element is dV rdrd dz
Theorem: (Change of coordinates)
Let E be the region:
Then the triple integral of f over E in cylindrical coordinates is
( ) ( cos , sin )
( ) ( cos , sin )( , , ) ( cos , sin , )
dV
f H r r
g G r rE
rdf x y z dV f z dr r z dr
( , , ) , ( ) ( ), ( , ) ( , )E r z g r h G x y z H x y
Triple Integrals in cylindrical coordinates – Example 2
Evaluate where E lies above z = 0, below z = y and inside the
cylinder . E
yzdV
sin23 2
0 00
sin2
r
zr drd
3 4 30 0
1 sin2
r drd
35
30
0
1 sin52
r d
53
0
3 sin10
d
1625
The plane in cylindrical coordinates is
The domain D is the semicircle of radius 3:
The integrand function in cylindrical coordinates is
0
sin3
0 0( sin )
E
r
dV
yzdV rdz drdr z
Triple Integrals in Cylindrical Coordinates – Example 3
Find the volume of the cone for using cylindrical coordinates.
2 2 2 24 16 x y x y
The cone and the plane intersect in a circle:
This circle defines the boundaries for and : ,
4 2 4
0 0
r
E
V dV rdzd dr
In cylindrical coordinates the cone has equation , thus
4
02 (4 ) r r dr
43
2
0
2 23
rr 643
4 2
0 0(4 )
r r d dr
Triple Integrals in Cylindrical Coordinates – Example 4
Sketch the solid whose volume is given by the integral and evaluate the integral. 2/2 2 9
0 0 0
r
rdzdrd
20 9 z r The solid is bounded below by (the -plane) and above by the paraboloid
0 2 r
The solid is bounded by the cylinder ( in cartesian: )
0 /2
The solid is in the first octant ( and )
22 /2 9
0 0 0
rrdzd dr
2 /2 20 0
(9 )
r r d dr
2 20
(9 )2 r r dr
2( 9 , 2 ) u r du rdr
5
94 udu 7