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