Triple Integrals in Cylindrical and Spherical Coordinates

Ryan C. Daileda

Trinity University

Calculus III

Daileda Cylindrical and Spherical Coordinates Introduction

As with double integrals, it can be useful to introduce other 3D coordinate systems to facilitate the evaluation of triple integrals.

We will primarily be interested in two particularly useful coordinate systems: cylindrical and spherical coordinates.

Cylindrical coordinates are closely connected to polar coordinates, which we have already studied.

Spherical coordinates, however, are a truly “new” coordinate system, so we will spend more time studying them.

Daileda Cylindrical and Spherical Coordinates Cylindrical Coordinates

Given a point P R3 with rectangular coordinates (x, y, z), its ∈ cylindrical coordinates (r, θ, z) are defined by:

P x = r cos θ y = r sin θ z z = z r 2 = x2 + y 2 r dV = dx dy dz = rdzdr dθ θ x y That is, we use the usual z-coordinate, but work in polar coordinates in the xy- Daileda Cylindrical and Spherical Coordinates Examples

Cylindrical coordinates are useful for describing regions E R3 ⊂ whose xy-projections have “nice” polar descriptions.

Example 1

Evaluate ez dV where E is enclosed by z =1+ x2 + y 2, E x2 + y 2 =ZZZ 5 and the xy-plane.

Solution. The region E is Type 1 with “bottom” z = 0, “top” z =1+ x2 + y 2, and xy-projection x2 + y 2 5. ≤ See Maple diagram.

Daileda Cylindrical and Spherical Coordinates Using cylindrical coordinates we have

2π √5 1+r 2 ez dV = ez rdzdrdθ ZZZE Z0 Z0 Z0 2π √5 1+r 2 = dθ ez r dz dr × Z0 Z0 Z0 √5 z=1+r 2 √5 2 = 2π ez r dr = 2π re1+r r dr − Z0 z=0 Z0

√5 2 = π e1+r r 2 = π(e6 e 5) . − − − 0 !

Daileda Cylindrical and Spherical Coordinates Example 2

Evaluate x dV where E is enclosed by z = 0, z = x + y + 5, ZZZE x2 + y 2 = 4 and x2 + y 2 = 9.

Solution. The region E is the portion of the tube centered on the z-axis, with inner 2, outer radius 3, between the planes z = 0 (the “bottom”) and z = x + y + 5 (the “top”).

See Maple diagram.

This is a Type 1 region, with cylindrical description 0 z r cos θ + r sin θ + 5, 2 r 3 and 0 θ 2π. ≤ ≤ ≤ ≤ ≤ ≤

Daileda Cylindrical and Spherical Coordinates Therefore 2π 3 r cos θ+r sin θ+5 x dV = r cos θ rdzdrdθ ZZZE Z0 Z2 Z0 2π 3 = r 2 cos θ (r cos θ + r sin θ + 5) dr dθ Z0 Z2 2π 3 = r 3 cos2 θ + r 3 sin θ cos θ + 5r 2 cos θ dr dθ Z0 Z2 3 2π 1 + cos 2θ = r 3 + r 3 sin θ cos θ + 5r 2 cos θ dθ dr 2 Z2 Z0   3 θ sin2θ r 2 sin2 θ θ=2π = r 3 + + + 5r 2 sin θ dr 2 4 2 Z2   θ=0

3 r 4 3 65π = π r 3 dr = π = . 4 4 Z2 2

Daileda Cylindrical and Spherical Coordinates Spherical Coordinates

Given a point P R3, its spherical coordinates (ρ, θ, φ) are given ∈ by:

P To describe all of R3 we need: 0 ρ< , ≤ ∞ z 0 θ 2π, ≤ ≤ 0 φ π. x ≤ ≤

y

Daileda Cylindrical and Spherical Coordinates Examples

Describe the following surfaces (k is a constant).

1. ρ = k: of radius k, centered at (0, 0, 0).

2. θ = k: Vertical half-plane “attached to” the z-axis.

3. φ = k: Cone with vertex at (0, 0, 0). Note:

The cone opens upward if 0 <φ<π/2.

The cone opens downward if π/2 <φ<π.

The cone is actually the xy-plane if φ = π/2.

Daileda Cylindrical and Spherical Coordinates More Examples

Describe the following regions.

1. ρ 2: Solid of radius 2, centered at (0, 0, 0). ≤ 2. ρ 2, 0 φ π/2: “Northern hemisphere” of preceding ball. ≤ ≤ ≤ 3. 2 ρ 3: Spherical shell or “cored” ball, with inner radius 2, ≤ ≤ outer radius 3, and center (0, 0, 0).

4. 2 ρ 3, 0 θ π/2, 0 φ π/2: Portion of the cored ≤ ≤ ≤ ≤ ≤ ≤ ball above in the first octant.

5. ρ 3, 0 φ π/4: Cone with a spherical “cap” of radius 3. ≤ ≤ ≤ 6. ρ 3, π/4 φ 3π/4: Solid ball with two cones removed. ≤ ≤ ≤

Daileda Cylindrical and Spherical Coordinates Relation to Rectangular Coordinates

We see that:

P x = r cos θ = ρ sin φ cos θ y = r sin θ z = ρ sin φ sin θ z = ρ cos φ x ρ2 = r 2 + z2 2 2 2 y = x + y + z .

To compute integrals in spherical coordinates, we need to relate dV = dx dy dz to dρ dφ dθ.

Daileda Cylindrical and Spherical Coordinates Change of Variables in Triple Integrals

If E, E R3 are connected by a transformation ′ ⊂ T (u, v, w) = (x(u, v, w), y(u, v, w), z(u, v, w))

(i.e. T : E E), one can use a Riemann sum argument to derive ′ → the relationship

∂(x, y, z) f (x, y, z) dV = f (T (u, v, w)) dudv dw, ′ ∂(u, v, w) ZZZE ZZZE

where ∂x/∂u ∂x/∂v ∂x/∂w ∂(x, y, z) = ∂y/∂u ∂y/∂v ∂y/∂w ∂(u, v, w) ∂z/∂u ∂z/∂v ∂z/∂w

is the Jacobian of the transformation.

Daileda Cylindrical and Spherical Coordinates The Jacobian of Spherical Coordinates

For the spherical coordinate transformation

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ,

we have sin φ cos θ ρ sin φ sin θ ρ cos φ cos θ ∂(x, y, z) − = sin φ sin θ ρ sin φ cos θ ρ cos φ sin θ = ρ2 sin φ, ∂(ρ, θ, φ) − cos φ 0 ρ sin φ −

so that dV = dx dy dz = ρ2 sin φ dρ dφ dθ .

Daileda Cylindrical and Spherical Coordinates Examples

Example 3

2 2 2 Evaluate e√x +y +z dV where E is the portion of the ball E x2 + y 2 +ZZZz2 9 in the first octant. ≤ Solution. In spherical coordinates we have

π/2 π/2 3 2 2 2 e√x +y +z dV = eρρ2 sin φ dρ dφ dθ ZZZE Z0 Z0 Z0 π/2 π/2 3 = dθ sin φ dφ ρ2eρ dρ × × Z0 Z0 Z0 π π/2 3 = cos φ (ρ2 2ρ + 2)eρ 2 − − 0 ! 0!

Daileda Cylindrical and Spherical Coordinates π π(5e3 2) = 1 5e3 2 = − . 2 · · − 2 

Example 4

Evaluate y dV , where E is bounded by the xz-plane and the ZZZE hemispheres y = √9 x2 z2 and y = √16 x2 z2. − − − −

Solution. E is the region between the x2 + y 2 + z2 = 9 and x2 + y 2 + z2 = 16 with y 0, i.e. the “right” half of a cored ball. ≥ This region would be quite difficult to express in rectangular coordinates, but is easy to describe spherically.

Daileda Cylindrical and Spherical Coordinates In spherical coordinates we have

π π 4 y dV = ρ sin φ sin θρ2 sin φ dρ dφ dθ ZZZE Z0 Z0 Z3 π π 4 = sin θ dθ sin2 φ dφ ρ3 dρ × × Z0 Z0 Z3 π π 1 cos 2φ ρ4 4 = cos θ − dφ − 2 4  0  Z0 3!

φ sin2 φ π 256 81 175 π = 2 − = . 2 − 4 4 4  0  

Daileda Cylindrical and Spherical Coordinates Example 5 Find the of the spherical “cap” of height h of a sphere of radius R.

Solution. Position the sphere at the origin with the cap at the north pole.

Let φ0 denote the angle that the edge of the cap makes with the north pole.

Daileda Cylindrical and Spherical Coordinates The volume of the cap is then the volume of the spherical region E described by 0 ρ R and 0 φ φ0, minus the volume of its ≤ ≤ ≤ ≤ lower conical portion (which can be computed from elementary ).

We have

2π φ0 R Vol(E)= dV = ρ2 sin φ dρ dφ dθ ZZZE Z0 Z0 Z0 2π φ0 R = dθ sin φ dφ ρ2 dρ × × Z0 Z0 Z0 φ0 ρ3 R 2πR3 = 2π cos φ = (1 cos φ0) . − 3 3 − 0 ! 0 !

Daileda Cylindrical and Spherical Coordinates Looking at a cross section through the north pole we find that

R h h cos φ0 = − = 1 . R − R

Thus 2πR3 2πR2h Vol(E)= (1 cos φ0)= . 3 − 3

Daileda Cylindrical and Spherical Coordinates The same diagram shows that the radius of the cone satisfies

r 2 + (R h)2 = R2 r 2 = R2 (R h)2 = 2Rh h2, − ⇒ − − − so that its volume is given by

πr 2(R h) π(2Rh h2)(R h) − = − − . 3 3

It now follows that the volume of the cap is

2πR2h π(2Rh h2)(R h) πh2 − − = (3R h) . 3 − 3 3 −

Daileda Cylindrical and Spherical Coordinates