CHATPER 15 - MULTIPLE INTEGRALS 15.1 Double Integrals Over Rectangles Review 5A – The development of the definite integral

Extend to Multivariable: Simple Case: Domain is a rectangular region. R: [a,b]X[c,d] DOMAIN R

HowdoweCALCULATEthisintegral?Forstarters,wecanESTIMATEitthewayweestimatedsingleintegralsusingaRiemannSum.Example:Estimate whereR=[0,2]X[0,2]issubdividedinto4subrectangleorequalsize,andchoosing

thesamplepointtobetheupperrightcornerofeachsubrectangle.

(16− x2 − 2y2 )dAR∫∫

Arethereapplicationsofdoubleintegralgeometricallyorphysically?Fromthelastexample,ifwetakemoreandmoresubrectangles,thisiswhatitwouldlooklike.

Usingthevolumeinterpretationtocomputeanintegral.Compute whereR=[0,3]X[0,1]

Otherphysicalapplicationsmass,area:

(2− 2y)dAR∫∫

CalculatingDoubleIntegralsasaniteratedintegral.Example:Calculatethevolumeunderz=4-x-yoverR:[0,2]X[0,1]Sketch:UseVolumebySlicing(5A:5.2)Case1:TakeslicesperpendiculartoX-AXIS

Case2:TakeslicesperpendiculartoY-AXIS

V = A(x)dx0

2

∫

V = A( y)dy0

1

∫

Example:Calculate whereR:[0,2]X[-1,1]

Example:Sometimes,choiceofordermatters:Calculate whereR:[1,2]X[0,p]

1− 6x2 y( )R∫∫ dA

ysin(xy)dAR∫∫

Review of 15.1

Last time, we defined the double integral of f(x,y) over a simple, rectangular region.

Whatdoesitmean? Iff(x,y)>0thenthedoubleintegralgivesthevolumeunderf(x,y)overR. Iff(x,y)=1,thenthedoubleintegralgivestheareaR. If hasphysicalmeaning(likemassperunitareatimesarea)thenthedoubleintegralisthetotalofthat physicalquantity(liketotalmassofR)Howdowecomputeit?

FromHW15.1

Toolfordoubleintegralsinspecialcasethatf(x,y)canbewrittenasg(x)h(y)…caution

15.2 Double Integrals Over General Regions 5Areviewproblem:Findtheareabetweeny=-xandy=x2over[0,1]TYPE1:dxTYPE2:dyGivenf(x,y)definedoveranon-rectangularregionD:

IfDisTYPE1Region

Example:Evaluate whereDisgivenasshown

Seedoubleintegralexampleon5Cpage:https://www.geogebra.org/m/ypbjEFuv

xy2 dAD∫∫

IfDisTYPE2Region:

Example: whereDisshownbelow.

(x + y)dD∫∫ A

Redopreviousexample,withDasaTYPE2region: whereDisgivenasshown

Theorderwechoosetointegratedependson___________________________________and__________________________.Sometimesthereisaclearlybetterchoice.

xy2 dAD∫∫

Example:Evaluate whereDistheregionenclosedby ,andthexaxis.

Example:Changingtheorderofintegration–recreatingthedomain.Evaluate:

ycos(x2 )dAD∫∫ y = x , x = π

2

ex2

dx dyy/2

1

∫0

2

∫

UsingdoubleintegralsasAREA:Usedoubleintegralstofindtheareabetweeny=-xandy=x2over[0,1]Often,thisideaisusedbackwards:Compute whereDistheregioncontainedin

4dAD∫∫ x2 + y2 = 25

15.3DoubleIntegralsInPolarCoordinatesMotivation:Evaluate: whereDisacircleofradius2,centeredattheorigin.

RecallPolarCoordinates:10.3:

DevelopmentofDoubleIntegralofSimpleRegion-“PolarRectangle”Given definedoverregionR=

Whatis ?

x2 + y2 dAD∫∫

x = r cosθy = r sinθ

x2 + y2 = r 2

tanθ = yx

(r,θ ) : a ≤ r ≤ b, α ≤θ ≤ β{ }

ΔAij

Whatis ?

Example: whereDistheregioninthefirstquadrantbetween and

ΔAij

x2 + y2 = 1 x2 + y2 = 5

Example:Evaluate: whereDisacircleofradius2,centeredattheorigin.

Extendingtheconceptofdoubleintegraltoamorecomplicatedpolarregion:

Example:Findthevolumeofthesolidthatliesunderthecone ,abovethexyplaneandinsidethecylinder

.

x2 + y2 dAD∫∫

z = x2 + y2

x2 + y2 = 2y

Example:UseadoubleintegraltofindtheAREAoftheregionenclosedby

r = 1− sinθ

15.6 Triple Integrals

Extend to Multivariable: , defined over some solid E in R3. Simple Case: Domain is a rectangular box. B: [a,b ] X [c,d] X [r,s] Partition [a,b] into l subintervals of equal width Partition [c,d] into m subintervals of equal width Partition [r,s] into n subintervals of equal width Consider typical “sub-box” Choose arbitrary point in sub-box: Form product: Sum over all sub-boxes.

Where

Application:Iff(x,y,z)=1, givethevolume.

If hasphysicalmeaning, givestotal…..

From5CpageunderTypesofIntegrals:

Soiff>0,Areacanbecomputedby:____________________________or__________________________________andvolumecanbecomputedby_________________________________or____________________________________________

Wecomputetripleintegralsasaniteratedintegral.Note:ThereareSIXpossibleordersofevaluation.

Example: forB=[0,1]X[0,p/2]X[0,3]

TripleIntegralsovernon-rectangularsolids:Recall “type 1 region” vs type 2 region in R2

Also recall various function orientations in R3

Given: , defined over some solid E in R3.

Example#1:Evaluate whereEisthesolidboundby2x+3y+z=6andthecoordinateplanes.

z = f (x, y, z)

2xE∫∫∫ dV

Example#2:Findthevolumeofthesolidboundby

SeeGeogebraAnimationhttps://www.geogebra.org/m/akme6U7F Recall:Wecanalsofindvolumeusingdoubleintegrals.

y = x2 , z = 0, y + z = 4

OtherorientationsforSolidE:

RedoExample#1fromdifferentorientationsEvaluate whereEisthesolidboundby2x+3y+z=6andthecoordinateplanes.

Theorderwechoosetointegratedependson__________________________-and___________________________

2xE∫∫∫ dV

RedoExample#2fromdifferentorientationsFindthevolumeofthesolidboundby

y = x2 , z = 0, y + z = 4

Example#3:

Whatdotheprojectionslooklike?THINKaboutit…..whichorderofintegrationmightbeeaser?Harder?

Example#3cont’d

Example#3cont’d

Example:

15.7 Triple Integrals in Cylindrical Coordinates Example(Leadinto15.7)Findthevolumeinsideofthecone

CylindricalCoordinates(especiallyusefulforcircularcylindersandcones)

Example: Switch the following integral to Cylindrical Coordinates

Cylindrical coordinates from other orientations. RevisitExample#3fromtheprevioussection:

15.8 Triple Integrals in Spherical Coordinates SphericalCoordinates(especiallyusefulforspheresandcones)ThepointPcanbeexressedas(r, q, f)where: r:distancefromtheorigintoP q:asbefore f:theanglebetweenthepositivezaxisandthelinesegmentOP. See on 5C page- Spherical Coordiate Animations https://mathinsight.org/spherical_coordinates Basic Graphs: r=ro

f=fo See on 5C page- Simple spherical solids https://www.geogebra.org/m/RtISr7GW#material/P4Avqxdr Derivation of Conversion Equations:

Example:ConvertingCoordinatesofPointsConvertfromRectangulartoSpherical ConvertfromSphericaltoRectangular

Examples:ConvertingEquationsConverttheequation tosphericalcoordinates.Converttheequation torectangularcoordinates.

z = x2 + y2

ρ = 2cosφ

DevelopmentofTripleIntegralinSphericalCoordinates.SimpleSphericalwedge:

Example:Evaluate whereEistherighthalfofthehemisphereofradius5.

Findthevolumeofthesolidboundby and

See5Cpage,animationof“snowcone”https://www.geogebra.org/m/tZgrSxQ4#material/xRQ2NMMk

(2− z)dVE∫∫∫

z = x2 + y2 x2 + y2 + z2 = z

RevisitPreviousExample:Findthevolumeinsideofthecone

z = 3x2 + 3y2 ; 0 ≤ z ≤ 3

16.2i and 16.7i Two more types of integrals

So far… New…

f(x)overinterval[a,b]

f(x,y)overregionD ____________f(x,y)over____________

f(x,y,z)oversolidE ____________f(x,y,z)over__________

f(x,y,z)oversolidE ____________f(x,y,z)over__________

16.2i Line (Contour) Integrals

Developmentoflineintegralforf(x,y)in (developmentforf(x,y,z)in is similar).

Given f(x, y) defined over some domain D and let C, given by , be a smooth ( conts and ) curve in D.

Partition [a,b] into n subintervals of equal Let Pi be the point on C corresponding to . These points break the curve into “sub-arcs”. Consider typical “sub-arc”, having length ___________ Choose arbitrary point in sub-arc: Form product: Sum over all sub-arcs.

R2 R3

!r (t) = x(t), y(t) ; a ≤ t ≤ b

′r (t) ′r (t) ≠ 0

Δt

!r (ti )

Where

What does this mean? Geometric: If f(x,y)>0 (examples from 5C page) Physical: If f=1, How do we compute it? We need to get in terms of t. In Math 5B (Section 8.1 and 10.2) we learned that

So or

AndinR3

Δsi

ds = dxdt

⎛⎝⎜

⎞⎠⎟

2

+ dydt

⎛⎝⎜

⎞⎠⎟

2

dt ds = !′r (t) dt

ds = dxdt

⎛⎝⎜

⎞⎠⎟

2

+ dydt

⎛⎝⎜

⎞⎠⎟

2

+ dzdt

⎛⎝⎜

⎞⎠⎟

2

dt

Andwecomputethelineintegral(ofascalarfunctionwithrespecttoarclength)byputtingitallintermsoft.

Example:Compute whereCisgivenby Example:Compute whereCisthelinesegmentfrom(1,0,4)to(-3,1,5).

f (x, y)ds = f (x(t), y(t))a

b

∫C∫

dxdt

⎛⎝⎜

⎞⎠⎟

2

+ dydt

⎛⎝⎜

⎞⎠⎟

2

dt

!r (t) = cost, sin t ; 0 ≤ t ≤ 2π

xy2zC∫ ds

Otherlineintegrals:Lineintegralswithrespecttox,y,combined

Example:Compute and whereCisgivenby

Oftenlineintegralsofthistypeoccurtogether:Andarewrittenintheform:

xy dxC∫ x2 dy

C∫

!r (t) = t, t2 ; 0 ≤ t ≤ 3

16.7iSurfaceIntegralsofscalarfunctionoversurfacegivenbyaFUNCTIONwithoutparametricsurfaces.NOTE:DONOTFOLLOWTHEBOOK’SAPPROACHHERE(NORTHEONLINESOLUTIONS).THEONLYPARTOFTHESECTIONTHATWEAREDOINGATTHISTIMEISONPAGE1165.

Given f(x, y, z) defined over some domain E and let S, given by z=g(x,y) over some domain

D be a surface contained in E.

Whatis ?

ΔSij

Finding ,theareaoftheijthpatch.(Seesection15.5)

Then wheredAisgivenby

Meaning:

Geometriciff(x,y,z)=1then

Physical:

ΔSij

f (x, y, z)dS = f (x, y,g(x, y))D∫∫

S∫∫ gx

2 + gy2 +1 dA

Example:Find whereSisthepartoftheparaboloid thatliesunderz=4.

OtherOrientations:ForsurfaceSgivenbyx=g(y,z)overaregionDintheyzplane:

wheredAcanbeviewedasdzdy,dydzorrdrdqExample:Find whereSistheportionofx=2-3y+z2thatliesoverthetriangleintheyzplanewithvertices

(0,0,0),(0,0,2)and(0,-4,2).

(Ans: )

f (x, y, z)dS = f (g( y, z), y, zD∫∫

S∫∫ ) gy

2 + gz2 +1 dA

(x + 3y − z2 ) dSS∫∫

263/2 −103/2

3

ForsurfaceSgivenbyy=g(x,z)overaregionDinthexzplane:

wheredAcanbeviewedasdzdx,dxdzorrdrdq

Example:Find whereSisthepartofthecylinder thatliesinthefirstoctantbetweenx=0andx=4.

(ans:36+12p)

y = 9− z2

Examplefrombookwithoutparametricsurfaces:PiecewiseSmoothSurface:

SeethebookforS2andS3.ForsurfaceS1,wewouldhavetobreakitintotwopieces, .S1a:

S1b:

y = ± 1− x2

y = + 1− x2

y = g(x, z) = 1− x2 ⇒ gx =−x

1− x2; gz = 0

⇒ dS = −x

1− x2⎛

⎝⎜⎞

⎠⎟

2

+ 02 +1 dA =!= 1

1− x2dA

z dS =S1a

∫∫D∫∫ z

1

1− x2dA

y = − 1− x2

y = g(x, z) = − 1− x2 ⇒ gx =x

1− x2; gz = 0

⇒ dS = x

1− x2⎛

⎝⎜⎞

⎠⎟

2

+ 02 +1 dA =!= 1

1− x2dA

z dS =S1b

∫∫D∫∫ z

1

1− x2dA

SurfaceAreaIff(x,y,z)=1thenthesurfaceintegralgivesussurfacearea.Thatis,theareaofthesurfacehavingprojectionDisgivenby:

SurfaceArea=

Example:Findtheareaofthepartofthehyperbolicparaboloid thatliesbetweenthecylinders and

x2 + y2 = 1x2 + y2 = 4