Section 16.8

Triple Integrals in Cylindrical and Spherical Coordinates

Recall that Cartesian and Cylindrical coordinates are related by the formulas

x = r cos θ, y = r sin θ, x2 + y2 = r2.

As a result, the function f (x, y, z) transforms into

f (x, y, z) = f (r cos θ, r sin θ, z) = F(r, θ, z).

CYLINDRICAL COORDINATES

TRIPLE INTEGRATION WITH CYLINDRICAL COORDINATES

Let E be a type 1 region and suppose that its projection D in the xy-plane can be described by

D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ θ ≤ h2(θ)}

If f is continuous, then

E

h

h

rru

rruddrdzrzrrfdVzyxf

)(

)(

)sin,cos(

)sin,cos(

2

1

2

1

),sin,cos(),,(

NOTE: The dz dy dx of Cartesian coordinates becomes r dz dr dθ in cylindrical coordinates.

EXAMPLES

1. Find the mass of the ellipsoid E given by 4x2 + 4y2 + z2 = 16, lying above the xy-plane. Then density at a point in the solid is proportional to the distance between the point and the xy-plane.

2. Evaluate the integral

2

2

4

4

22

2 22

x

x yxdxdydzx

The equations that relate spherical coordinates to Cartesian coordinates are

SPHERICAL COORDINATES AND SPHERICAL WEDGES

cos

,sinsin,cossin

z

yx

In spherical coordinates, the counterpart of a rectangular box is a spherical wedge

dcbaE ,,|),,(

Divide E into smaller spherical wedges Eijk by means of equally spaced spheres ρ = ρi, half-planes θ = θj, and half-cones φ = φk. Each Eijk is approximately a rectangular box with dimensions Δρ, ρiΔφ, and ρi sin φk Δθ. So, the approximate volume of Eijk is given by

Then the triple integral over E can be given by the Riemann sum

kikiiijkV sinsin 2

pointsampleais~

,~

,~),,(where

~sin~)

~cos~,

~cos

~sin~,

~cos

~sin~(lim

),,(lim),,(

***

1 1 1

2

,,

1 1 1

***

,,

kjiijkijkijk

l

i

m

j

n

kkjikikijkijki

nml

l

i

m

j

n

kijkijkijkijk

Enml

zyx

f

VzyxfdVzyxf

The Riemann sum on the previous slide gives us

where E is a spherical wedge given by

TRIPLE INTEGRATION IN SPHERICAL COORDINATES

dcbaE ,,|),,(

d

c

b

a

E

dddf

dVzyxf

sincos,sinsin,cossin

),,(

2

EXTENSION OF THE FORMULA

d

c

g

g

E

dddf

dVzyxf

),(

),(

22

1

sincos,sinsin,cossin

),,(

The formula can be extended to included more general spherical regions such as

The triple integral would become

),(),(,,|),,( 21 ggdcE

1. Use spherical coordinates to evaluate the integral

2. Find the volume of the solid region E bounded

by below by the cone and above by the sphere x2 + y2 + z2 = 3z.

EXAMPLES

dydxdzzyxy yx

yx

3

0

9

0

18 2222 22

22)(

22 yxz