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15.7 Triple Integrals in Cylindrical and Spherical Coordinates1

Chapter 15.Multiple Integrals

15.7.Triple Integrals in Cylindrical and Spherical

Coordinates

Definition.Cylindrical coordinatesrepresent a pointPin space by ordered triples (r,θ,z) in which

1.randθare polar coordinates for the vertical projection ofPon the

xy-plane

2.zis the rectangular vertical coordinate.

Figure 15.42, Page 893

15.7 Triple Integrals in Cylindrical and Spherical Coordinates2

Note.The equations relating rectangular (x,y,z) and cylindrical (t,θ,z) coordinates are x=rcosθ, y=rsinθ, z=z r

2=x2+y2,tanθ=y/x.

Note.In cylindrical coordinates, the equationr=adescribes not just a circle in thexy-plane but an entire cylinder about thez-axis. Thez- axis is given byr= 0. The equationθ=θ0describes the plane that contains thez-axis and makes an angleθ0with the positivex-axis. And, just as in rectangular coordinates, the equationz=z0describes a plane perpendicular to thez-axis.

Figure 15.43, Page 894

15.7 Triple Integrals in Cylindrical and Spherical Coordinates3

Note.When computing triple integrals over a regionDin cylindrical coordinates, we partition the region intonsmall cylindrical wedges, rather than into rectangular boxes. In thekth cylindrical wedge,r,θandz change by Δrk, Δθk, and Δzk, and the largest of these numbers among all the cylindrical wedges is called thenormof the partition. We define the triple integral as a limit of Riemann sums using these wedges. Thee volume of such a cylindrical wedge ΔVkis obtained by taking the area ΔAkof its base in therθ-plane and multiplying by the height Δz. For a point (rk,θk,zk) in the center of thekth wedge, we calculated in polar coordinates that ΔAk=rkΔrkΔθk. So ΔVk= ΔzkrkΔrkΔθkand a

Riemann sum forfoverDhas the form

S n=n? The triple integral of a functionfoverDis obtained by taking a limit of such Riemann sums with partitions whose norms approach zero: lim ?P?→0Sn=? ? ? D f dV=? ? ? D f dz rdr dθ.

15.7 Triple Integrals in Cylindrical and Spherical Coordinates4

Figure 15.44, Page 894

Example.Page 901, number 4.

How to Integrate in Cylindrical Coordinates

To evaluate

D f(r,θ,z)dVover a regionDin space in cylindrical coordinates, integrating first with respect toz, then with respect tor, and finally with respect toθ, take the following steps.

1.Sketch.Sketch the regionDalong with its projectionRon thexy-

15.7 Triple Integrals in Cylindrical and Spherical Coordinates5

plane. Label the surfaces and curves that boundDandR.

Page 895

2.Find thez-limits of integration.Draw a lineMpassing through a

typical point (r,θ) ofRparallel to thez-axis. Aszincreases,M entersDatz=g1(r,θ) and leaves atz=g2(r,θ). These are the z-limits of integration.

Page 895

15.7 Triple Integrals in Cylindrical and Spherical Coordinates6

3.Find ther-limits of integration.Draw a rayLthrough (r,θ) from

the origin. The ray entersRatr=h1(θ) and leaves atr=h2(θ).

These are ther-limits of integration.

Page 896

4.Find theθ-limits of integration.AsLsweeps acrossR, the angleθ

it makes with the positivex-axis runs fromθ=αtoθ=β. These are theθ-limits if integration. The integral is D f(r,θ,z)dV=? r=h2(θ) r=h1(θ)? z=g2(r,θ) z=g1(r,θ)f(r,θ,z)dzr dr dθ.

Example.Page 902, number 18.

15.7 Triple Integrals in Cylindrical and Spherical Coordinates7

Definition.Spherical coordinatesrepresent a pointPin space by or- dered triples (ρ,φ,θ) in which

1.ρis the distance fromPto the origin (notice thatρ >0).

2.φis the angle?OPmakes with the positivez-axis (φ?[0,π]).

3.θis the angle from cylindrical coordinate (θ?[0,2π]).

Figure 15.47, Page 897

15.7 Triple Integrals in Cylindrical and Spherical Coordinates8

Note.The equationρ=adescribes the sphere of radiusacentered at the origin. The equationφ=φ0describes a single cone whose vertex lies at the origin and whose axis lies along thez-axis.

Figure 15.48, Page 897

Note.The equations relating spherical coordinates to Cartesiancoordi- nates and cylindrical coordinates are r=ρsinθ, x=rcosθ=ρsinφcosθ, z=ρcosφ, y=rsinθ=ρsinφsinθ, x2+y2+z2=⎷r2+z2.

15.7 Triple Integrals in Cylindrical and Spherical Coordinates9

Note.When computing triple integrals over a regionDin spherical coordinates, we partition the region intonspherical wedges. The size of thekth spherical wedge, which contains a point (ρk,φk,θk), is given be the changes Δρk, Δθk, and Δφkinρ,θ, andφ. Such a spherical wedge has one edge a circular arc of lengthρkΔφk, another edge a circular arc of lengthρksinφkΔθk, and thickness Δρk. The spherical wedge closely approximates a cube of these dimensions when Δρk, Δθk, and Δφkare all small. It can be shown that the volume of this spherical wedge ΔVk is ΔVk=ρ2ksinφkΔρkΔφkΔθkfor (ρk,φk,θk) a point chosen inside the wedge. The corresponding Riemann sum for a functionf(ρ,φ,θ) is S n=n? As the norm of a partition approaches zero, and the sphericalwedges get smaller, the Riemann sums have a limit whenfis continuous: lim ?P?→0Sn=? ? ? D f(ρ,φ,θ)dV=? ? ? D

15.7 Triple Integrals in Cylindrical and Spherical Coordinates10

Figure 15.51, Page 898

Example.Page 902, number 26.

How to Integrate in Spherical Coordinates

To evaluate

D f(ρ,φ,θ)dVover a regionDin space in spherical coordinates, integrating first with respect toρ, then with respect toφ, and finally with respect toθ, take the following steps.

1.Sketch.Sketch the regionDalong with its projectionRon thexy-

15.7 Triple Integrals in Cylindrical and Spherical Coordinates11

plane. Label the surfaces and curves that boundDandR.

Page 899

2.Find theρ-limits of integration.Draw a rayMfrom the origin

throughDmaking an angleφwith the positivez-axis. Also draw the projection ofMon thexy-plane (call the projectionL). The rayL makes an angleθwith the positivex-axis. Asρincreases,Menters Datρ=g1(φ,θ) and leaves atρ=g2(φ,θ). These are theρ-limits

15.7 Triple Integrals in Cylindrical and Spherical Coordinates12

of integration.

Page 899

3.Find theφ-limits of integration.For any givenθ, the angleφthat

Mmakes with thez-axis runs fromφ=φmintoφ=φmax. These are theφ-limits of integration.

4.Find theθ-limits of integration.The rayLsweeps overRasθruns

fromαtoβ. These are theθ-limits of integration. The integral is

Df(ρ,φ,θ)dV=?

θ=α?φ=φmax

φ=φmin?ρ=g2(φ,θ)

ρ=g1(φ,θ)

f(ρ,φ,θ)ρ2sinφdρdφ dθ.

Example.Page 903, number 34.

15.7 Triple Integrals in Cylindrical and Spherical Coordinates13

Note.In summary, we have the following relationships.

Cylindrical to Spherical to Spherical to

Rectangular Rectangular Cylindrical

x=rcosθ x=ρsinφcosθ r=ρsinφ y=rsinθ y=ρsinφsinθ z=ρcosφ z=z z=ρcosθ θ=θ

In terms of the differential of volume, we have

dV=dxdy dz=dz r dr dθ=ρ2sinφdρdφdθ. Examples.Page 903, number 46. Page 904, number 54.quotesdbs_dbs20.pdfusesText_26