25 oct 2019 · Its polar coordinate equation is r = 2 sinθ P Sam Johnson Triple Integrals in Cylindrical and Spherical Coordinates October 25, 2019 13/67
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25 oct 2019 · Its polar coordinate equation is r = 2 sinθ P Sam Johnson Triple Integrals in Cylindrical and Spherical Coordinates October 25, 2019 13/67
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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 1/67Overview
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 2/67Integration 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 theform (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 which1.randare polar coordinates for the vertical projection ofPon the
xy-plane2.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; r2=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 5/67Cylindrical 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 6/67Cylindrical 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 thearea 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 : SketchTo 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 thesurfaces 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 IntegrationDraw 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 ExampleExample 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 coordinateequation 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 0f(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 x2+y2= 4;bounded above by the paraboloidz=x2+y2;and bounded
below by thexyplane.Solution :We sketch the solid, bounded above by the paraboloidz=r2and 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 angleitmakes 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 52dr 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]r20r dr d
Z 2 0Z 2 0 r3dr d=Z 2 0 r44 2 0 d=Z 2 04d= 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 17/67Spherical 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