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Triple Integrals in Cylindrical and Spherical Coordinates

P. Sam Johnson

October 25, 2019

P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 1/67

Overview

When a calculation in physics, engineering, or geometry involves a cylinder, cone, sphere, we can often simplify our work by using cylindrical or spherical coordinates, which are introduced in the lecture. The procedure for transforming to these coordinates and evaluating the resulting triple integrals is similar to the transformation to polar coordinates in the plane discussed earlier. P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 2/67

Integration in Cylindrical Coordinates

We obtain cylindrical coordinates for space by combining polar coordinates in thexy-plane with the usualz-axis. This assigns to every point in space one or more coordinate triples of the

form (r;;z).P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 3/67

Integration in Cylindrical Coordinates

Denition 1.

Cylindrical coordinatesrepresent a pointPin space by ordered triples (r;;z)in which

1.randare polar coordinates for the vertical projection ofPon the

xy-plane

2.zis the rectangular vertical coordinate.The values ofx;y;r;andin rectangular and cylindrical coordinates are

related by the usual equations. Equations Relating Rectangular(x;y;z)and Cylindrical(r;;z)

Coordinates :

x=rcos;y=rsin;z=z; r

2=x2+y2;tan=y=x:P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 4/67

Constant-coordinate Equations in Cylindrical Coordinates 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 equation=0describes the plane

that contains thez-axis and makes an an- gle0with the positivex-axis.

And, just as in rectangular coordinates,

the equationz=z0describes a plane per- pendicular to thez-axis.

Thus constant-coordinate equations in

cylindrical coordinates yield cylinders and planes. P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 5/67

Cylindrical Coordinates

Cylindrical coordinates are good for describing cylinders whose axes run along thez-axis and planes the either contain thez-axis or lie perpendicular to thez-axis. Surfaces like these have equations of constant coordinate values: r= 4 Cylinder, radius 4, axis thez-axis ==3 Plane containing thez-axis z= 2 Plane perpendicular to thez-axis When computing triple integrals over a regionDin cylindrical coordinates, we partition the region intonsmall cylindrical wedges, rather than into rectangular boxes. P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 6/67

Cylindrical Coordinates

In thekth cylindrical wedge,r;andzchange by rk;k;and zk, and the largest of these numbers among all the cylindrical wedges is called the normof the partition. We dene the triple integral as a limit of Riemann sums using these wedges. The volume of such a cylindrical wedge Vkis obtained by taking the

area Akof its base in ther-plane and multiplying by the height z.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 7/67

Cylindrical Coordinates

For a point (rk;k;zk) in the center of thekth wedge, we calculated in polar coordinates that Ak=rkrkk. So Vk= zkrkrkk and a Riemann sum forfoverDhas the form S n=nX k=1f(rk;k;zk)zkrkrkk: The triple integral of a functionfoverDis obtained by taking a limit of such Riemann sums with partitions whose norms approach zero lim n!1=ZZZ D f dV=ZZZ D f dz r dr d: Triple integrals in cylindrical coordinates are then evaluated as iterated integrals. P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 8/67 How to Integrate in Cylindrical Coordinates : Sketch

To evaluate

ZZZ D f(r;;z)dV over a regionDin space in cylindrical coordinates, integrating rst with respect toz, then with respect tor, and nally with respect to, take the following steps. Sketch the regionDalong with its projectionRon thexy-plane. Label the

surfaces and curves that boundDandR.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 9/67

How to Integrate in Cylindrical Coordinates : Thez-Limits of Integration Draw a lineMthrough a typical point (r;) ofRparallel to thez-axis. Aszincreases,MentersDatz=g1(r;) and leaves atz=g2(r;).

These are thez-limts of integration.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 10/67

How to Integrate in Cylindrical Coordinates : Ther-Limits of Integration

Draw a rayLthrough (r;) from the origin.

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

These are ther-limits of integration.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 11/67

How to Integrate in Cylindrical Coordinates : The-Limits of Integration AsLsweeps acrossR, the angleit makes with the positivex-axis runs from=to=.

These are the-limits of integration.

The integral is

ZZZ D f(r;;z)dV=Z =Z r=h2() r=h1()Z z=g2(r;)

z=g1(r;)f(r;;z)dz r dr d:P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 12/67

How to Integrate in Cylindrical Coordinates - An Example

Example 2.

Letf(r;;z)be a function dened over the regionDbounded below by the planez= 0, laterally by the circular cylinderx2+ (y1)2= 1, and above by the paraboloidz=x2+y2:The base ofDis also the region's projectionRon thexy-plane. The boundary ofRis the circlex2+ (y1)2= 1. Its polar coordinate

equation isr= 2sin.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 13/67

How to Integrate in Cylindrical Coordinates - An Example We nd the limits of integration, starting with thez-limits. A lineM through typical point (r;) inRparallel to thez-axis entersDatz= 0 and leaves atz=x2+y2=r2. Next we nd ther-limits of integration. A rayLthrough (r;) from the origin entersRatr= 0 and leaves atr= 2sin. Finally we nd the-limits of integration. AsLsweeps acrossR, the angle it makes with the positivex-axis runs from= 0 to=.

The integral is

ZZZ D f(r;;z)dV=Z 0Z 2sin 0Z r2 0

f(r;;z)dz r dr d:P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 14/67

Example

Example 3.

Find the centroid(= 1)of the solid enclosed by the cylinder x

2+y2= 4;bounded above by the paraboloidz=x2+y2;and bounded

below by thexyplane.Solution :We sketch the solid, bounded above by the paraboloidz=r2

and below by the planez= 0.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 15/67

Solution (contd...)

Its baseRis the disk 0r2 in thexyplane. The solid's centroid (x;y;z) lies on its axis of symmetry, here the z-axis. This makes x= y= 0. To nd z, we divide the rst momentMxyby the massM. To nd the limits of integration for the mass and moment integrals, we continue with the four basic steps. We completed our initial sketch. The remaining steps give the limits of integration. Thez-limits. A lineMthrough a typical point (r;) in the base parallel to the z-axis enters the solid atz= 0 and leaves atz=r2: Therlimits. A rayLthrough (r;) from the origin entersRatr= 0 and leaves atr= 2: Thelimits:AsLsweeps over the base like a clock hand, the angleit

makes with the positive x-axis runs from= 0 to= 2.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 16/67

Solution (contd...)

The value ofMxyis

M xy=Z 2 0Z 2 0Z r2 0 z dz r dr d=Z 2 0Z 2 0 z22 r2 0 r dr d Z 2 0Z 2 0r 52
dr d=Z 2 0 r612 2 0 d=Z 2 0163
d=323

The value ofMis

M=Z 2 0Z 2 0Z r2 0 dz r dr d=Z 2 0Z 2 0 [z]r2

0r dr d

Z 2 0Z 2 0 r3dr d=Z 2 0 r44 2 0 d=Z 2 0

4d= 8:

Therefore z=MxyM

=323 18=43 ;and the centroid is (0;0;4=3). Notice that the centroid lies outside the solid. P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 17/67

Spherical Coordinates and Integration

Spherical coordinates locate points in space with two angles and one distance. The rst coordinate,=j!OPj, is the point's distance from the origin.

Unliker, the variableis never negative.P. Sam JohnsonTriple Integrals in Cylindrical and Spherical CoordinatesOctober 25, 2019 18/67

Spherical Coordinates

The scond coordinate,, is the anglej!OPjmakes with the positivez-axis.

It is required to lie in the interval [0;].

The third coordinate is the angleas measured in cylindrical coordinates.Denition 4. Spherical Coordinatesrepresent a pointPin space by ordered triplesquotesdbs_dbs20.pdfusesText_26

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