Preliminaries:

slope1: defines forward light cone, extending from slope1 to slope2. 
We make sure that slope1 -> 0<= θ1 <= π
                                        slope2 -> 0<= θ2-θ1 
So you have to make sure the slopes are entered in the correct order in order to mark the timelike separated events

lightCone[x0_,y0_,len_,slope1_,slope2_,color_]:= Module[
{x1,y1,x2,y2,x3,y3,x4,y4,θ1,θ2,θ, cone,l},
l=Abs[len];
If[slope1>0,
θ1=ArcTan[slope1],
θ1=ArcTan[slope1]+π
];(* ArcTan gives  ((-π)/2) <= θ <= (π/2)*)
If[slope2>0,
θ2=ArcTan[slope2],
θ2=ArcTan[slope2]+π
];
If[θ2<θ1,θ=θ2;θ2=θ1;θ1=θ];
x1=x0+l Cos[θ1];y1= y0+ l Sin[θ1];
x2=x0+l Cos[θ2];y2= y0+ l Sin[θ2];
x3=x0-l Cos[θ2];y3=y0 - l Sin[θ2];
x4=x0-l Cos[θ1];y4=y0 - l Sin[θ1];
cone=Polygon[{{x1,y1},{x2,y2},{x0,y0},{x4,y4},{x3,y3},{x0,y0}}];
(*Print["P1= (",x1,",",y1,")  P2= (",x2,",",y2,")"];*)
Graphics[{color,cone}]
];
(*Show[{lightCone[0.,0.,1.,-1.,-4.5,Red],lightCone[2.,3.,1.,1.,3.6,Blue]}]*)

Problem 5

v1[x_,v0_] := v0;
v2[x_,x0_]:= Log[((((x/x0)))^2)];
dv1[x_] := 0;
dv2[x_]:= (2/x);

xmin=-5;xmax=5;vmin=-5;vmax=3.5;
v0s={-3,-2,-1,0,1,2,3};
x0s={0.25,1,3,7};
g1=Plot[Table[v1[x,v0],{v0,v0s}],{x,xmin,xmax}];
g2=Plot[Table[v2[x,x0],{x0,x0s}],{x,xmin,xmax}];
vp=-1;xp=x/.FindRoot[v2[x,7]==vp,{x,4}];
l1=lightCone[xp,vp,1,dv1[xp],dv2[xp],Magenta];
vp=-1;xp=x/.FindRoot[v2[x,3]==vp,{x,2}];
l2=lightCone[xp,vp,0.8,dv1[xp],dv2[xp],Magenta];
vp=-1;xp=x/.FindRoot[v2[x,1]==vp,{x,0.5}];
l3=lightCone[xp,vp,0.7,dv1[xp],dv2[xp],Magenta];
vp=-1;xp=x/.FindRoot[v2[x,1]==vp,{x,-0.5}];
l4=lightCone[xp,vp,0.7,dv1[xp],dv2[xp],Magenta];
vp=-1;xp=x/.FindRoot[v2[x,3]==vp,{x,-2}];
l5=lightCone[xp,vp,0.8,dv1[xp],dv2[xp],Magenta];
vp=-1;xp=x/.FindRoot[v2[x,7]==vp,{x,-4}];
l6=lightCone[xp,vp,1,dv1[xp],dv2[xp],Magenta];
Print["(x,v)= (",xp,",",vp,")"];
Show[g1,g2,l1,l2,l3,l4,l5,l6,
PlotRange->{{xmin,xmax},{vmin,vmax}}, AspectRatio-> 1, Axes-> True,AxesLabel->{"x","v"}]

Problem 18

Integrate[(1/(Sqrt[1-2(M/r)])),r,Assumptions->M>0 && r>2M]

Integrate[(1/(Sqrt[1-2(M/r)])),{r,2M,3M},Assumptions->M>0 && r>2M]

N[%]

i1 = Integrate[((r^2)/(Sqrt[1-2(M/r)])),r,Assumptions->M>0 && r>2M]

Integrate[((r^2)/(Sqrt[1-2(M/r)])),{r,2M,3M},Assumptions->M>0 && r>2M]

N[%]

4. π %

Problem 20

Integrate[(Sqrt[2M])(((r-2M))^(-1/2)),r,Assumptions->M>0 && r>2M]

RevolutionPlot3D[2(Sqrt[2(r-2)]),{r,2.1,5},Axes-> False,Boxed->False]