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