538 CHAPTER 27. THE INTEGRAL IN OTHER COORDINATES

18. A right circular cone has a base of radius 2 and a height equal to 2. Use polarcoordinates to find its volume.

19. Now suppose in the above problem, it is not really a cone but instead z = 2− 12 r2.

Find its volume.

27.3 Cylindrical and Spherical CoordinatesCylindrical coordinates are defined as follows.

x(r,θ ,z) ≡

 xyz

=

 r cos(θ)r sin(θ)

z

 ,

r ≥ 0,θ ∈ [0,2π),z ∈ R

Spherical coordinates are a little harder. These are given by

x(ρ,θ ,φ) ≡

 xyz

=

 ρ sin(φ)cos(θ)ρ sin(φ)sin(θ)

ρ cos(φ)

 ,

ρ ≥ 0,θ ∈ [0,2π),φ ∈ [0,π]

The following picture relates the various coordinates.

x1 (x1,y1,0)

y1

(ρ,φ ,θ)(r,θ ,z1)(x1,y1,z1)

z1

ρ

rθ

φ

•

x

y

z

In this picture, ρ is the distance between the origin, the point whose Cartesian coor-dinates are (0,0,0) and the point indicated by a dot and labelled as (x1,y1,z1), (r,θ ,z1),and (ρ,φ ,θ). The angle between the positive z axis and the line between the origin andthe point indicated by a dot is denoted by φ , and θ is the angle between the positive xaxis and the line joining the origin to the point (x1,y1,0) as shown, while r is the lengthof this line. Thus r = ρ sin(φ) and is the usual polar coordinate while θ is the other polarcoordinate. Letting z1 denote the usual z coordinate of a point in three dimensions, likethe one shown as a dot, (r,θ ,z1) are the cylindrical coordinates of the dotted point. Thespherical coordinates are determined by (ρ,φ ,θ). When ρ is specified, this indicates thatthe point of interest is on some sphere of radius ρ which is centered at the origin. Thenwhen φ is given, the location of the point is narrowed down to a circle of “latitude” andfinally, θ determines which point is on this circle by specifying a circle of “longitude”. Letφ ∈ [0,π],θ ∈ [0,2π), and ρ ∈ [0,∞). The picture shows how to relate these new coordinatesystems to Cartesian coordinates. Note that θ is the same in the two coordinate systemsand that ρ sinφ = r.