SF1626 - Övning 9

Göran Andersson goeran@kth.se, 110427

7.8

a)

Integrationsområdet är enhetssfären.

𝒦={(x,y,z):(x^2)+(y^2)+(z^2)≤1}

Rymdpolära koordinater:

(Piecewise[{{x=r sin(θ)cos(ϕ)}, {y=r sin(θ)sin(ϕ)}, {z=r cos(θ)}}])=>((d(x,y,z))/(d(r,θ,ϕ)))=(Det[({{sin(θ)cos(ϕ), r cos(θ)cos(ϕ), r sin(θ)(-sin(ϕ))}, {sin(θ)sin(ϕ), r cos(θ)sin(ϕ), r sin(θ)cos(ϕ)}, {cos(θ), r (-sin(θ)), 0}})])=(r^2) sin(θ)

=>(Abs[((d(x,y,z))/(d(r,θ,ϕ)))])=(Abs[(r^2) sin(θ)])(Overscript[=, 0≤θ≤π])(r^2) sin(θ)

=>𝒦'={(r,θ,ϕ):0≤r≤1,0≤θ≤π,0≤ϕ<2π}

(Underscript[∫∫∫, 𝒦])((x^2)+(y^2)+(z^2))dx dy dz=(Underscript[∫∫∫, 𝒦'])(r^2)((r^2) sin(θ))dr dθ dϕ

=(∫_0^1) (r^4)dr(∫_0^π) sin(θ)dθ(∫_0^(2π)) dϕ=(Subscript[[(1/5)(r^2)], 0]^1) ·(Subscript[[-cos(θ)], 0]^π)·(Subscript[[ϕ], 0]^(2π))

=(1/5)·2·2π=((4π)/5)

b)

(Underscript[∫∫∫, 𝒦])(x^2)dx dy dz(Overscript[=, symmetri])(1/3)(Underscript[∫∫∫, 𝒦])((x^2)+(y^2)+(z^2))dx dy dz=((4π)/15)

Alternativ 1:

(Underscript[∫∫∫, 𝒦])(x^2)dx dy dz=(Underscript[∫∫∫, 𝒦'])(((r sin(θ)cos(ϕ)))^2)((r^2) sin(θ))dr dθ dϕ

=(∫_0^1) (r^4)dr(∫_0^π) (sin^2)(θ)sin(θ)dθ(∫_0^(2π)) (cos^2)(ϕ)dϕ=(1/5)(∫_0^π) (1-(cos^2)(θ))sin(θ)dθ(∫_0^(2π)) (1/2) (1+cos(2 ϕ))dϕ

=(1/5)·(∫_1^(-1)) (1-(t^2))(-dt) ·π=((4π)/15)

Alternativ 2:

(Underscript[∫∫∫, 𝒦])(x^2) dx dy dz=(∫_(-1)^1)(x^2) ((Underscript[∫∫, (y^2)+(z^2)≤1-(x^2)]) dy dz)dx=(∫_(-1)^1)(Underscript[(Underscript[(x^2)  π (1-(x^2)), ⏟]), jämn])dx=2π(∫_0^1)((x )^2) (1-(x^2))dx=((4π)/15)

c)

(Underscript[∫∫∫, 𝒦])x dx dy dz=(∫_(-1)^1)x ((Underscript[∫∫, (y^2)+(z^2)≤1-(x^2)]) dy dz)dx=(∫_(-1)^1)(Underscript[(Underscript[x  π (1-(x^2)), ⏟]), udda])dx=0

7.11

Integrationsområdet är

𝒟={(x,y,z):(x^2)+(y^2)+(z^2)≤1∧y≥0∧z≥0}

Rymdpolära koordinater införs:

=>𝒟'={(r,θ,ϕ):0≤r≤1,0≤θ≤(π/2),0≤ϕ≤π}

(Underscript[∫∫∫, 𝒟])z dx dy dz=(Underscript[∫∫∫, 𝒟'])(r cos(θ))((r^2) sin(θ))dr dθ dϕ

=(∫_0^1) (r^3)dr(∫_0^(π/2)) cos(θ)sin(θ)dθ(∫_0^π) dϕ=(1/4)·(1/2)·π=(π/8)

7.13

Integrationsområdet är

𝒟={(x,y,z):(x^2)+(y^2)+(z^2)≤1∧z≥(Sqrt[(x^2)+(y^2)])}={(x,y,z):(Sqrt[(x^2)+(y^2)])≤z≤(Sqrt[1-(x^2)-(y^2)])}

För z≥0 får vi skärningen mellan sfär och kon:

(Piecewise[{{(x^2)+(y^2)+(z^2)=1}, {(x^2)+(y^2)=(z^2)}}])⇔(Piecewise[{{(x^2)+(y^2)=(1/2)}, {z=(1/(Sqrt[2]))}}])

Med rymdpolära koordinater blir skärningen r=1,θ=π/4,0≤ϕ≤2π. Det nya integrationsområdet blir därmed

=>𝒟'={(r,θ,ϕ):0≤r≤1,0≤θ≤(π/4),0≤ϕ<2π}

(Underscript[∫∫∫, 𝒟])(Sqrt[(x^2)+(y^2)+(z^2)]) dx dy dz=(Underscript[∫∫∫, 𝒟'])r ((r^2) sin(θ))dr dθ dϕ

=(∫_0^1) (r^3)dr(∫_0^(π/4)) sin(θ)dθ(∫_0^π) dϕ=(1/4)·(1-(1/(Sqrt[2])))·2π=(π/2) (1-(1/(Sqrt[2])))

7.14

Integrationsområdet är

𝒟={(x,y,z):(x^2)+(y^2)+(z^2)≥1}

Rymdpolära koordinater införs

=>𝒟'={(r,θ,ϕ):r≥1,0≤θ≤π,0≤ϕ≤2π}

(Underscript[∫∫∫, 𝒟])(1/((((x^2)+(y^2)+(z^2)))^(α/2))) dx dy dz=(Underscript[∫∫∫, 𝒟'])(1/(r^α))((r^2) sin(θ))dr dθ dϕ

=(∫_1^∞) (r^(2-α))dr(∫_0^π) sin(θ)dθ(∫_0^(2π)) dϕ(Overscript[=, 2-α<-1])(Subscript[[((r^(3-α))/(3-α))], r=1]^∞) ·2·2π

=(0-(1/(3-α)))·2·2π=((4π)/(α-3))

Integralen (∫_1^∞) (x^a)dx är konvergent om a<-1. Detta motsvarar här 2-α<-1⇔α>3.

7.15

Rymdpolära koordinater införs

=>𝒟'={(r,θ,ϕ):0≤r<∞,0≤θ≤π,0≤ϕ≤2π}

(Underscript[∫∫∫, (ℝ^2)])((e^(-(Sqrt[(x^2)+(y^2)+(z^2)])))/(Sqrt[(x^2)+(y^2)+(z^2)])) dx dy dz=(Underscript[∫∫∫, 𝒟'])((e^(-r))/r)((r^2) sin(θ))dr dθ dϕ

=(∫_0^∞) r (e^(-r))dr(∫_0^π) sin(θ)dθ(∫_0^(2π)) dϕ=((Subscript[[r  (-(e^(-r)))], r=0]^∞)-(∫_0^∞) (-(e^(-r)))dr)·2·2π

=4π(∫_0^∞) (e^(-r))dr=4π·1=4π

7.17

Inför koordinaterna:

(Piecewise[{{u=x+2 y+2 z}, {v=2 x-2 y+z}, {w=2 x+y-2 z}}])=>((d(u,v,w))/(d(x,y,z)))=(Det[({{1, 2, 2}, {2, -2, 1}, {2, 1, -2}})])=27=>((d(x,y,z))/(d(u,v,w)))=(1/27)

∴(Underscript[∫∫∫, (ℝ^3)])((e^(-(((x+2 y+2 z))^2)))/((1+(((2 x-2 y+z))^2)) (1+(((2 x+y-2 z))^2))))dx dy dz=(Underscript[∫∫∫, (ℝ^3)])((e^(-(u^2)))/((1+(v^2)) (1+(w^2))))(1/27)du dv dw

=(1/27)(∫_(-∞)^∞) (e^(-(u^2)))du(∫_(-∞)^∞) (1/(1+(v^2)))dv(∫_(-∞)^∞) (1/(1+(w^2)))dw=(1/27)(Sqrt[π]) ((((∫_(-∞)^∞) (1/(1+(v^2)))dv))^2)

=(1/27)(Sqrt[π]) ((((Subscript[[arctan(v)], v=-∞]^∞)))^2)=(1/27)(Sqrt[π])(π^2)=((π^(5/2))/27)

8.2

V=(Underscript[∫∫∫, 𝒦]) dx dy dz=(Underscript[∫∫, 𝒟]) ((∫_((x^2)+(y^2))^(Sqrt[2-((x^2)+(y^2))])) dz )dx dy=(Underscript[∫∫, 𝒟])((Sqrt[2-((x^2)+(y^2))])-((x^2)+(y^2)))dx dy

Ytorna skär varandra i

(Piecewise[{{(z^2)=2-((x^2)+(y^2))}, {z=(x^2)+(y^2)}}])=>(Piecewise[{{z=1}, {(x^2)+(y^2)=1}}])

Projektonen i xy-planet blir:

𝒟={(x,y):(x^2)+(y^2)≤1}

polära koordinater=>𝒟'={(r,θ),∶0≤r≤1∧0≤θ≤2π}

V=(Underscript[∫∫, 𝒟'])((Sqrt[2-(r^2)])-(r^2))r dr dθ=(∫_0^1)((Sqrt[2-(r^2)])-(r^2)) r dr(∫_0^(2π)) dθ

=2π(∫_0^1)((Sqrt[2-(r^2)])-(r^2)) r dr =2π(∫_0^1)(Sqrt[2-(r^2)]) r dr (Piecewise[{{t=2-(r^2)}, {dt=(-2 r) dr}}])-2π(∫_0^1)(r^3) dr

=π(∫_1^2)(Sqrt[t]) dt -2π(1/4)=π(2/3) (2 (Sqrt[2])-1)-2π(1/4)=(1/6) (8 (Sqrt[2])-7) π

8.4

Integrationsområdet är en hyperboloid, utskuren ur en sfär med radien 7.

V=(Underscript[∫∫∫, 𝒦]) dx dy dz=(Underscript[∫∫, 𝒟]) ((∫_(Sqrt[1+(x^2)+(y^2)])^(Sqrt[(7^2)-((x^2)+(y^2))])) dz )dx dy=(Underscript[∫∫, 𝒟])((Sqrt[(7^2)-((x^2)+(y^2))])-(Sqrt[1+(x^2)+(y^2)]))dx dy

Ytorna skär varandra i

(Piecewise[{{(x^2)+(y^2)+(z^2)=(7^2)}, {(x^2)+(y^2)-(z^2)=1}}])⇔(x^2)+(y^2)=24

Projektonen i xy-planet blir:

𝒟={(x,y):(x^2)+(y^2)≤24}

polära koordinater=>𝒟'={(r,θ),∶0≤r≤(Sqrt[24])∧0≤θ≤2π}

∴V=(Underscript[∫∫, 𝒟'])((Sqrt[(7^2)-(r^2)])-(Sqrt[1+(r^2)]))r dr dθ=(∫_0^(Sqrt[24]))((Sqrt[(7^2)-(r^2)])-(Sqrt[1+(r^2)])) r dr(∫_0^(2π)) dθ

=2π(∫_0^(Sqrt[24]))((Sqrt[(7^2)-(r^2)])-(Sqrt[1+(r^2)])) r dr (Piecewise[{{t=(r^2)}, {dt=2 r dr}}])=π(∫_0^24)((Sqrt[(7^2)-t])-(Sqrt[1+t])) dt

=(Subscript[π  [-(((((7^2)-t))^(3/2))/(3/2))-((((1+t))^(3/2))/(3/2))], t=0]^24)==((2π)/3) ((-125-125)-(-343-1))=((188π)/3)

Alternativ: cylindriska koordinater

(Piecewise[{{x=r cos(θ)}, {y=r sin(θ)}, {z=z}}])=>((d(x,y,z))/(d(r,θ,z)))=(Det[({{cos(θ), -r sin(θ), 0}, {sin(θ), r cos(θ), 0}, {0, 0, 1}})])=r

𝒟={(x,y,z):(x^2)+(y^2)+(z^2)≤(7^2),z≥(Sqrt[1+(x^2)+(y^2)])}

=>𝒟'={(r,θ,z):0≤r≤(Sqrt[24]),0≤θ≤2π,(Sqrt[1+(r^2)])≤z≤(Sqrt[(7^2)-(r^2)])}

Volymen blir

V=(Underscript[∫∫∫, 𝒟])dx dy dz=(Underscript[∫∫∫, 𝒟'])(Abs[((d(x,y,z))/(d(r,θ,z)))])dr dθ dz=(Underscript[∫∫∫, 𝒟'])r dr dθ dz

=(∫_0^(2π)) dθ(∫_0^(Sqrt[24]))r ((∫_(Sqrt[1+(r^2)])^(Sqrt[(7^2)-(r^2)]))dz) dr=π(∫_0^(Sqrt[24]))((Sqrt[(7^2)-(r^2)])-(Sqrt[1+(r^2)]))2r dr(Piecewise[{{t=(r^2)}, {dt=2r dr}}])

=π(∫_0^24) ((Sqrt[(7^2)-t])-(Sqrt[1+t]) ) dt=(Subscript[π  [-(((((7^2)-t))^(3/2))/(3/2))-((((1+t))^(3/2))/(3/2))], t=0]^24)

=((2π)/3) ((-125-125)-(-343-1))=((188π)/3)

8.8

Integrationsområdet 𝒦 utgörs av skärningen mellan två paraboliska cylindrar z≤1-(((x/2))^2) och z≤1-(((y/2))^2) för z≥0. För fixt z utgörs snittarean av en kvadrat med arean (2x)(2 y)=4(x^2)=16(1-z).

V=(Underscript[∫∫∫, 𝒦])dx dy dz=(∫_0^1)((Underscript[∫∫, 𝒟])dx dy )dz=(∫_0^1)16(1-z) dz=8

8.11

Integrationsområdet 𝒦 utgörs av utrymmet mellan två ytor (x^2)+(y^2)-1≤z≤(Sqrt[(x^2)+(y^2)])+1.

V=(Underscript[∫∫∫, 𝒦]) dx dy dz=(Underscript[∫∫, 𝒟]) ((∫_((x^2)+(y^2)-1)^((Sqrt[(x^2)+(y^2)])+1)) dz )dx dy=(Underscript[∫∫, 𝒟])((Sqrt[(x^2)+(y^2)])-((x^2)+(y^2))+2))dx dy

Ytorna skär varandra i

(Piecewise[{{z=(Sqrt[(x^2)+(y^2)])+1}, {z=(x^2)+(y^2)-1}}])⇔(Piecewise[{{z=3}, {(x^2)+(y^2)=4}}])

Projektonen i xy-planet blir:

𝒟={(x,y):(x^2)+(y^2)≤4}

polära koordinater=>𝒟'={(r,θ),∶0≤r≤2∧0≤θ≤2π}

∴V=(Underscript[∫∫, 𝒟'])(r-(r^2)+2)r dr dθ=(∫_0^2)((r^2)-(r^3)+2r)  dr(∫_0^(2π)) dθ =(8/3)·2π=((16π)/3)

8.13

Integrationsområdet är.

𝒦={(x,y,z):((((x/a)))^2)+((((y/b)))^2)+((((z/c)))^2)≤1}

Inför koordinaterna:

(Piecewise[{{u=x/a}, {v=y/b}, {w=z/c}}])⇔(Piecewise[{{x=a u}, {y=b v}, {z=c w}}])⇔((d(x,y,z))/(d(u,v,w)))=(Det[({{a, 0, 0}, {0, b, 0}, {0, 0, c}})])=a b c

𝒦'={(u,v,w):(u^2)+(v^2)+(w^2)≤1}

V=(Underscript[∫∫∫, 𝒦])dx dy dz=(Underscript[∫∫∫, 𝒦'])(Abs[((d(x,y,z))/(d(r,θ,ϕ)))])du dv dw=a b c (Underscript[∫∫∫, 𝒦'])du dv dw=a b c ((4π)/3)

8.28

V=(Underscript[∫∫∫, 𝒦])dx dy dz=(Underscript[∫∫, 𝒟])((∫_(Sqrt[(x^2)+(y^2)])^(Sqrt[1-((x^2)+(y^2))]))dz)dx dy =(Underscript[∫∫, 𝒟])((Sqrt[1-((x^2)+(y^2))])-(Sqrt[(x^2)+(y^2)]))dx dy

Ytorna skär varandra i

(Piecewise[{{(z^2)=1-((x^2)+(y^2))}, {(z^2)=(x^2)+(y^2)}}])⇔(x^2)+(y^2)=(1/2)

Denna cirkel begränsar integrationsområdet i xy-planet.

𝒟={(x,y):(x^2)+(y^2)≤(1/2)}

polära kordinater ger:

𝒟'={(r,θ):0≤r≤(1/(Sqrt[2])),0≤θ≤2π}

∴V=(Underscript[∫∫, 𝒟])((Sqrt[1-((x^2)+(y^2))])-(Sqrt[(x^2)+(y^2)]))dx dy  =(Underscript[∫∫, 𝒟'])((Sqrt[1-(r^2)])-r)r dr dθ

=(∫_0^(1/(Sqrt[2])))((Sqrt[1-(r^2)])-r) r dr(∫_0^(2π)) dθ =2π(∫_0^(1/(Sqrt[2])))(Sqrt[1-(r^2)]) r dr (Piecewise[{{t=1-(r^2)}, {dt=(-2 r) dr}}])-2π(∫_0^(1/(Sqrt[2])))(r^2) dr

=π(∫_(1/2)^1)(Sqrt[t]) dt -(π/(3 (Sqrt[2])))=(π/3) (2-(Sqrt[2]))

Kroppen är rotationssymmetrisk med avseende på z-axeln vilket innebär att (Subscript[x, mc])=(Subscript[y, mc])=0. För z-koordinaten erhålls:

(Underscript[∫∫∫, 𝒦])z dx dy dz=(Underscript[∫∫, 𝒟])((∫_(Sqrt[(x^2)+(y^2)])^(Sqrt[1-((x^2)+(y^2))]))z dz)dx dy =(Underscript[∫∫, 𝒟])((Subscript[[(1/2)(z^2)], z=(Sqrt[(x^2)+(y^2)])]^(Sqrt[1-((x^2)+(y^2))])))dx dy =(Underscript[∫∫, 𝒟])((1/2)-((x^2)+(y^2)))dx dy

=(∫_0^(1/(Sqrt[2])))((1/2)-(r^2)) r dr(∫_0^(2π)) dθ =(π/8)

∴(Subscript[z, mc])=(π/8)/((π/3) (2-(Sqrt[2]))) =(3/(8 (2-(Sqrt[2]))))=(3/16)(2+(Sqrt[2]))=>(Subscript[𝕩, mc])=(0,0,(3/16)(2+(Sqrt[2])))

8.31

a)

V=(Underscript[∫∫∫, 𝒦])dx dy dz=(Underscript[∫∫, 𝒟])((∫_(y^2)^(2-(x^2)-(y^2)))dz)dx dy =(Underscript[∫∫, 𝒟])(2-(x^2)-2(y^2))dx dy

Ytorna skär varandra i

(Piecewise[{{z=2-(x^2)-(y^2)}, {z=(y^2)}}])⇔(x^2)+2(y^2)=2

Denna kurva begränsar integrationsområdet

𝒟={(x,y):(x^2)+2(y^2)≤2}={(x,y):((((x/(Sqrt[2]))))^2)+(y^2)≤1}

Elliptiska kordinater ger:

(Piecewise[{{(x/(Sqrt[2]))=r cos(θ)}, {y=r sin(θ)}}])=>((d(x,y))/(d(r,θ)))=(Det[({{(Sqrt[2]) cos(θ), -r (Sqrt[2])sin(θ)}, {sin(θ), r cos(θ)}})])=(Sqrt[2])r

(Overscript[=>, r≥0])(Abs[((d(x,y))/(d(r,θ)))])=(Sqrt[2])r

𝒟'={(r,θ):0≤r≤1,0≤θ≤2π}

∴V=(Underscript[∫∫, 𝒟])(2-(x^2)-2(y^2))dx dy =2(Underscript[∫∫, 𝒟])(1-((((x/(Sqrt[2]))))^2)-(y^2))dx dy =

=2(Underscript[∫∫, 𝒟'])(1-(r^2))(Sqrt[2])r dr dθ =2(Sqrt[2])(∫_0^(2π))dθ(∫_0^1)(r-(r^3)) dr

=2(Sqrt[2])2π (1/4)=(Sqrt[2]) π

b)

Kroppen är helt symmetrisk m.a.p. x=0 och y=0, dvs. (Subscript[x, mc])=(Subscript[y, mc])=0.

(Subscript[z, mc])V=(Underscript[∫∫∫, 𝒦])z dx dy dz=(Underscript[∫∫, 𝒟])((∫_(y^2)^(2-(x^2)-(y^2)))z dz)dx dy =(1/2)(Underscript[∫∫, 𝒟]) ((((2-(x^2)-(y^2)))^2)-(y^4))dx dy

=(1/2)(Underscript[∫∫, 𝒟]) (2-(x^2)-2(y^2))(2-(x^2))dx dy =2(Underscript[∫∫, 𝒟]) (1-((((x/(Sqrt[2]))))^2)-(y^2))(1-((((x/(Sqrt[2]))))^2))dx dy

=2(Underscript[∫∫, 𝒟']) (1-(r^2)) (1-(r^2)(cos^2)(θ))(Sqrt[2])r dr dθ=2(Sqrt[2])(∫_0^(2π))((∫_0^1)(((r^5)-(r^3)) (cos^2)(θ)+r-(r^3))  dr )dθ

=2(Sqrt[2])(∫_0^(2π))((1/4)-(((cos^2)(θ))/12) )dθ=((Sqrt[2])/12)(∫_0^(2π))(5-cos(2 θ)) dθ=((Sqrt[2])/12)10 π

=((Sqrt[2])/2)((11 π)/12)=>(Subscript[z, mc])=(1/V)((Sqrt[2])/2)((11 π)/12)=

=((Sqrt[2])/16)(∫_0^(2π))(1-(1/6)(cos^2)(θ) )dθ=((Sqrt[2])/16)(∫_0^(2π))((11-cos(2 θ))/12)dθ=((Sqrt[2])/16)10π

(Subscript[z, mc])=((((Sqrt[2])/12)10 π)/((Sqrt[2]) π))=(5/6)

Dvs. masscentrum blir

(Subscript[𝕣, mc])=(0,0,(5/6))