Triple Integral in Spherical Coordinates

Spherical Coordinates

Spherical coordinates:

= angle of the projection of the point into the -plane measured from the positive x-axis ( )

Relationship with cartesian:

, ,

with

= distance |OP| ()

ϕ = angle |OP| makes with the positive z- axis (0 ≤ ϕ ≤ π; if ϕ > π/2 the point has a negative z-coordinate)

cossincos

xryr

z

sinr

Also: 2 2 2 2 2 2 r z x y z

sin cossin sincos

xyz

thus

tan yx

Spherical CoordinatesBasic graphs in spherical coordinates:

represents a sphere of radius ( in cartesian)

represents a vertical plane

ϕ = c represents a cone

Spherical Coordinates

Plot the point and convert to cartesian: 2, ,3 4

( , , )

Change from rectangular to spherical: 1,1, 2( , , ) x y z

24 3 2

64 3 2

24

2sin cos

2sin sin

2cos

x

y

z 2 6( , , ) , , 22 2

x y z

22 2 2 2 2 21 1 2 x y z22

cos cos = zz 4

=tan 1 y

xThe projection of the point in the -plane is in the first quadrant

4 =arctan(1)=

2, ,4 4

( , , )

is the equation of the bottom half of the cone.

Spherical Coordinates Example 1

Write the equation of the cone in spherical coordinates.

Recall that .sin r

Also, and .cos z

Substituting into the equation of the cone, yields:

2 2 2 2

2 2

cos sin z r

Simplifying: 2 2 2sincos tan 1 tan 1

Thus 3,

4 4

is the equation of the top half of the cone.4

34

Triple Integrals in spherical coordinates

The volume element is

2 sin ddV d d

Theorem: Change of coordinates

where E is the spherical wedge given by

( , , )E

f x y z dV2 ( sin cos , sin sin , c sin os )

d b

c a dVd d df

, , , , E a b c d

Triple Integrals in Spherical Coordinates - Example 4

Evaluate where E is the hemispherical region that lies above

the -plane and below the sphere

2 2E

x y dV

hemisphere

0 3

2

0 20 2 2

E

x y dV

2 3 4 320 0 0

sin

d d d

320

243(2 ) sin5

d

3245

2 2

2 32

0 0 022 2 sinsin

x y dV

d d d

Spherical coordinates:

2 320 0

243 sin5

d d

We need to determine the angle that describes the cone in spherical coordinates.

Triple Integrals in Spherical Coordinates - Example 5

Find the volume of the solid that lies within the sphere , above the -plane and outside the cone

2 24 z x y2 2sincos 4

cos 4 sin 1 1tan = =arctan4 4

2 7 220 0

sin

E dV

V dV d d d 3 22

0

7 sin3

d d

3272 sin

3

d

3 /272 cos3

372 cos

3

696.92

37 12 cos arctan3 4

Triple Integrals in Spherical Coordinates - Example 6

Evaluate the integral by changing to spherical coordinates2 2 2

2 2

1 1 2

0 0

x x y

x yxy dzdydx

2 2 2 22 x y z x y

The solid is bounded below by the cone and above by the hemisphere . The radius of the hemisphere is

20 10 1

y xx

The projection of the solid in the -plane is the quarter of the disk in the first quadrant.

Triple Integrals in Spherical Coordinates Example 6 continued

/2 /4 2 20 0 0

( sin cos )( sin sin ) sin

x y dV

d d d

In spherical coordinates:• the cone has equation • the sphere has equation

4

The limits of integration are

40

20

20

/2 /4 2 4 30 0 0

sin sin cos

d d d

5 /2 /4 30 0

( 2) sin sin cos5

d d 4 2 5 .044

15

2 2 2

2 2

1 1 2

0 0

x x y

x yxy dzdydx