Problem 61 in 16.3

?RegionPlot3D

RegionPlot3D[x^2+y^3-z^2>0,{x,-2,2},{y,-2,2},{z,-2,2}]

RegionPlot3D[And[x>0,x<1,y>0,y<1,z>0,z<1],{x,-2,2},{y,-2,2},{z,-2,2},PlotPoints->{100,100,100}]

RegionPlot3D[And[x≥0,y≥0,z ≥0,(x^2)+3(y^2)≤12,z<6 (y^2)],{x,0,2(Sqrt[3])},{y,0,2},{z,0,24},PlotPoints->{100,100,100}]

RegionPlot3D[And[x≥0,y≥0,z ≥0,y≤2,x≤(Sqrt[12-3 (y^2)]),z<6 (y^2)],{x,0,2(Sqrt[3])},{y,0,2},{z,0,24},PlotPoints->{100,100,100}]

The corresponding integral for the volume is:

Integrate[Integrate[Integrate[1,{x,0,(Sqrt[12-3 (y^2)])}],{z,0,6 y^2}],{y,0,2}]

This is not an easy integral since

Integrate[Integrate[1,{x,0,(Sqrt[12-3 (y^2)])}],{z,0,6 y^2}]

Integrate[6 (y^2) (Sqrt[12-3 (y^2)]),y]

Integrate[(y^2) (Sqrt[4-(y^2)]),y]

FullSimplify[D[(1/4) y (Sqrt[4-(y^2)]) (-2+(y^2))+2 ArcSin[(y/2)],y]]

RegionPlot3D[And[x>0,x<2(Sqrt[3]),y>0,y<(Sqrt[4-((x^2)/3)]),z>0,z<6 (y^2)],{x,0,2(Sqrt[3])},{y,0,2},{z,0,24},PlotPoints->{100,100,100}]

The corresponding integral for the volume is:

Integrate[Integrate[Integrate[1,{z,0,6 y^2}],{y,0,(Sqrt[4-((x^2)/3)])}],{x,0,2(Sqrt[3])}]

This is not an easy integral since

Integrate[Integrate[1,{z,0,6 y^2}],{y,0,(Sqrt[4-((x^2)/3)])}]

Integrate[2 (((4-((x^2)/3)))^(3/2)),x]

RegionPlot3D[And[x>0,x<(Sqrt[12-(z/2)]),y>(Sqrt[(z/6)]),y<(Sqrt[4-((x^2)/3)]),z>0,z<24],{x,0,2(Sqrt[3])},{y,0,2},{z,0,24},PlotPoints->{100,100,100}]

The corresponding integral for the volume is:

Integrate[Integrate[Integrate[1,{y,(Sqrt[(z/6)]),(Sqrt[4-((x^2)/3)])}],{x,0,(Sqrt[12-(z/2)])}],{z,0,24}]

This is not an easy integral since

Integrate[Integrate[1,{y,(Sqrt[(z/6)]),(Sqrt[4-((x^2)/3)])}],{x,0,(Sqrt[12-(z/2)])},Assumptions-> And[z>0,z<24]]

Cutting the unit disk with parallel lines and the unit sphere with parallel planes

Solve[θ-(1/2)Sin[2 θ]==(π/3),θ]

FindRoot[θ-(1/2)Sin[2 θ]==(π/3),{θ,2Pi/3}]

Cos[1.3026628373004512`]

3Integrate[2(Sqrt[1-(x^2)]),{x,-0.264932084602777`,0.264932084602777`}]

Solve[(x^3)-3 x+(2/3)==0,x]//N

Solve[(x^3)-3 x+(2/3)==0,x]

FullSimplify[-((1-I (Sqrt[3]))/(2 ((((1/3) (-1+2 I (Sqrt[2]))))^(1/3))))-(1/2) ((((1/3) (-1+2 I (Sqrt[2]))))^(1/3)) (1+I (Sqrt[3]))]

N[%]

mx=0.226073713789208`

Integrate[2 (Sqrt[1-(x^2)-(y^2)]),{y,-(Sqrt[1-(x^2)]),(Sqrt[1-(x^2)])}]

(9/4)Integrate[π-π (x^2),{x,-mx,mx}]