15.4 Homework-Double Integral Appns PDF

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10/31/2017

15.4 Homework-Double Integral Appns

WebAssign 15.4HomeworkDoubleIntegralAppns(Homework) CurrentScore:–/10

ClaireJenkins APMAE2000,section001,Fall2017 Instructor:DrewYoungren

Due:Saturday,October28201708:40AMEDT

Theduedateforthisassignmentispast.Yourworkcanbeviewedbelow,butnochangescanbemade.  Important!Beforeyouviewtheanswerkey,decidewhetherornotyouplantorequestanextension.YourInstructormaynotgrant youanextensionifyouhaveviewedtheanswerkey.Automaticextensionsarenotgrantedifyouhaveviewedtheanswerkey. RequestExtension

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1. –/1pointsSCalcET815.4.AE.003.

EXAMPLE3 Thedensityatanypointonasemicircularlamina isproportionaltothedistancefromthecenterofthecircle.Find thecenterofmassofthelamina.  SOLUTION Let'splacethelaminaastheupperhalfofthe circle x2+y2=a2. (Seethefigure.)Thenthedistancefroma  VideoExample 



point (x,y) tothecenterofthecircle(theorigin)is x2+y2 . Thereforethedensityfunctionis x2+y2

ρ(x,y)=K

whereKissomeconstant.Boththedensityfunctionandthe shapeofthelaminasuggestthatweconverttopolar x2+y2 =r andtheregionDisgivenby

coordinates.Then

0≤r≤a,  0≤θ≤π. Thusthemassofthelaminais 

ρ(x,y)dA

m = D



=

x2+y2 dA

K D a

π 0

0

π

a

r2drdθ

= K 0

=

rdrdθ

(NoResponse) 

=

0

a

Kπ (NoResponse) 

 0

= (NoResponse) 

.

Boththelaminaandthedensityfunctionaresymmetricwith respecttotheyaxis,sothecenterofmassmustlieonthe yaxis,thatis x= (NoResponse) 



0  . Theycoordinate

isgivenby y

=

=

=

=

1 m

yρ(x,y)dA D

Kπa3 0

3

rsin(θ)(Kr)rdrdθ 0

π

3

πa3

a

π

3

πa3 http://www.webassign.net/web/Student/Assignment-Responses/view_key?dep=16938807



a

sin(θ)dθ

(NoResponse) 

dr

0

0

π

−cos(θ)

a

(NoResponse)  0

 0

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= (NoResponse) 

.

Thecenterofmassislocatedatthepoint (x,y)= (NoResponse) 



 .



2. –/1pointsSCalcET815.4.AE.004.

EXAMPLE4 Findthemomentsofinertia Ix,Iy,andI0 ofa homogeneousdiskDwithdensity ρ(x,y)=ρ, centerthe origin,andradiusa.  SOLUTION TheboundaryofDisthecircle x2+y2=a2 and inpolarcoordinatesDisdescribedby 0≤θ≤2π,  0≤r≤a. Let'scomputeI0first: 

I0

(x2+y2)ρdA

= D

2π

=

a

r2rdrdθ

ρ 0

0 a

2π

=

ρ 0

=

r3dr

dθ 0

a

2πρ (NoResponse) 

 0

= (NoResponse) 

.

Insteadofcomputing IxandIy directly,weusethefactsthat Ix+Iy=I0 and Ix=Iy (fromthesymmetryoftheproblem). Thus

Ix=Iy=



I0 = (NoResponse)  2

.



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3. –/1pointsSCalcET815.4.AE.005.

EXAMPLE5 Findtheradiusofgyrationaboutthexaxisofa homogeneousdiskDwithdensity ρ(x,y)=ρ, centerthe origin,andradiusa.  SOLUTION Themassofthediskis m=ρπa2, sofromthese equationswehave 2= Ix =

m

(NoResponse) 

= (NoResponse)

ρπa2 . Thereforetheradiusofgyrationaboutthexaxisis

= (NoResponse) 



, whichishalftheradiusofthedisk.



4. –/1pointsSCalcET815.4.005.

FindthemassandcenterofmassofthelaminathatoccupiestheregionDandhasthegivendensity functionρ. Disthetriangularregionwithvertices(0,0),(2,1),(0,3);ρ(x,y)=8(x+y) m = (NoResponse)  (x,y) =

(NoResponse) 



SolutionorExplanation 2

2 1 2 y=3−x 3 1 1 y dx =8 x 3− x + (3−x)2− x2 dx 2 2 2 8 y=x/2 0 0 0 x/2 2 2 9 1 3 9 9 9 x + x =48, = 8 − x2+ dx =8 −  8 2 8 3 2 0 0 2

3−x

8(x+y)dydx =8

m =

2

xy+

2

2 1 2 y=3−x 9 9 xy dx =8 x− x3 dx =36,  2 2 8 y=x/2 0 0 0 x/2 2 2 2 3−x y=3−x 1 2 9 1 Mx= 9− x =72.  xy + y3 dx =8 8(xy+y2)dydx=8 2 3 2 y = x /2 0 0 0 x/2 My Mx 3 3 , = , . Hence m=48,(x,y)= 4 2 m m 3−x

My=

8(x2+xy)dydx=8



x2y+



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5. –/1pointsSCalcET815.4.011.

Alaminaoccupiesthepartofthedisk x2+y2≤25 inthefirstquadrant.Finditscenterofmassifthe densityatanypointisproportionaltoitsdistancefromthexaxis. (x,y)= (NoResponse) 

 

SolutionorExplanation 5

π/2

kr2sinθdrdθ =

ρ(x,y)=ky=krsinθ,m= 0

0

1 k 3 0

π/2

5

r3 sinθdθ = 0

125 k 3 0

π/2

sinθdθ =

π/2 125 125 k −cosθ =  k, 3 3 0 5

π/2

My=

kr3sinθcosθdrdθ = 0

0

1 k 4 0

π/2

5

r4 sinθcosθdθ= 0

625 k 4 0

π/2

sinθcosθdθ =

625 8

π/2

625 k −cos2θ = k, 8 0 5

π/2

kr3sin2θdrdθ =

Mx= 0

0

1 k 4 0

π/2

5

r4 sin2θdθ = 0

625 k 4 0

π/2

sin2θdθ =

625 1 k θ− sin2θ 8 2

π/2

625 = πk. 16 0 Hence(x,y)= 

15 15 , π . 8 16





6. –/1pointsSCalcET815.4.504.XP.

FindthemomentsofinertiaIx,Iy,I0foralaminathatoccupiesthepartofthediskx2+y2≤16inthefirst quadrantifthedensityatanypointisproportionaltothesquareofitsdistancefromtheorigin.(Assume thatthecoefficientofproportionalityisk.)

Ix= (NoResponse) 



Iy= (NoResponse) 



I0= (NoResponse) 



SolutionorExplanation ClicktoViewSolution





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7. –/1pointsSCalcET815.4.509.XP.MI.

Electricchargeisdistributedoverthediskx2+y2≤16sothatthechargedensityat(x,y)is

ρ(x,y)=7x+7y+7x2+7y2(measuredincoulombspersquaremeter).Findthetotalchargeonthedisk. (NoResponse) 

C

SolutionorExplanation ClicktoViewSolution







8. –/1pointsSCalcET815.4.023.

Alaminawithconstantdensity ρ(x,y)=ρ occupiesthegivenregion.Findthemomentsofinertia Ix and Iy andtheradiiofgyration and . Thepartofthedisk x2+y2≤a2 inthefirstquadrant Ix = (NoResponse)  Iy =

=

=

(NoResponse) 

(NoResponse) 

(NoResponse) 

SolutionorExplanation ClicktoViewSolution





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9. –/1pointsSCalcET815.4.027.

ThejointdensityfunctionforapairofrandomvariablesXandYisgiven.(Roundyouranswerstofour decimalplaces.) f(x,y)=

Cx(1+y) if0≤x≤3,0≤y≤2 0 otherwise

(a)FindthevalueoftheconstantC. (NoResponse) 



0.0556 

 (b)FindP(X≤1,Y≤1). (NoResponse) 



0.0417 

 (c)FindP(X+Y≤1). (NoResponse) 



0.0116

SolutionorExplanation ClicktoViewSolution







10.–/1pointsSCalcET815.4.508.XP.

Electricchargeisdistributedovertherectangle2≤x≤4,0≤y≤2sothatthechargedensityat(x,y)is

σ(x,y)=2xy+y2(measuredincoulombspersquaremeter).Findthetotalchargeontherectangle. (NoResponse) 

C

SolutionorExplanation ClicktoViewSolution







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