Problem #1:

2 Dimensions

Integrate[ r ,{r,0,R},{ϕ,0,2π}]

Integrate[ r DiracDelta[r-R],{r,0,Infinity},{ϕ,0,2π},Assumptions->{R>0}]

3 Dimensions

Integrate[ (r^2)Sin[θ] ,{r,0,R},{θ,0,π},{ϕ,0,2π}]

Integrate[ (r^2)Sin[θ]  DiracDelta[r-R],{r,0,Infinity},{θ,0,π},{ϕ,0,2π},Assumptions->{R>0}]

4 Dimensions

Integrate[ (r^3)Sin[θ] ((Sin[ψ])^2),{r,0,R},{θ,0,π},{ψ,0,π},{ϕ,0,2π}]

Integrate[ (r^3)Sin[θ] ((Sin[ψ])^2)  DiracDelta[r-R],{r,0,Infinity},{θ,0,π},{ψ,0,π},{ϕ,0,2π},Assumptions->{R>0}]

5 Dimensions

Integrate[ (r^3)Sin[θ] ((Sin[ψ])^2)((Sin[β])^3),{r,0,R},{θ,0,π},{ψ,0,π},{β,0,π},{ϕ,0,2π}]

Integrate[ (r^3)Sin[θ] ((Sin[ψ])^2) ((Sin[β])^3) DiracDelta[r-R],{r,0,Infinity},{θ,0,π},{ψ,0,π},{β,0,π},{ϕ,0,2π},Assumptions->{R>0}]

Problem #2:

Integrate[ (5 x - 2) DiracDelta[2-x],{x,0,3}]

Integrate[ ϕ[x]DiracDelta[(x^2)-(a^2)],{x,0,Infinity},Assumptions->{a>0}]

Integrate[ Cos[x]DiracDelta[-2 x ],{x,-1,1}]

Integrate[ Cos[x]DiracDelta[Sin[x] ],{x,((-π)/2),((+π)/2)}]

Problem #3: Warmup

Clear["Global`*"]

f[x_]= (1/2)DiracDelta[x-1]+(1/2)DiracDelta[x+1]

x f[x] //Expand

Integrate[ x f[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f[x],{x,-Infinity,Infinity}]

f2[y_]=Convolve[f[x],f[x],x,y] //Expand

Integrate[ x f2[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f2[x],{x,-Infinity,Infinity}]

f3[y_]=Convolve[f[x],f2[x],x,y] //Expand

Integrate[ x f3[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f3[x],{x,-Infinity,Infinity}]

f4[y_]=Convolve[f[x],f3[x],x,y] //Expand

Integrate[ x f4[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f4[x],{x,-Infinity,Infinity}]

Problem #3:

Clear["Global`*"]

f[x_]= (2/3)DiracDelta[x-1]+(1/3)DiracDelta[x+1]

x f[x] //Expand

Integrate[ x f[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f[x],{x,-Infinity,Infinity}]

f2[y_]=Convolve[f[x],f[x],x,y] //Expand

Integrate[ x f2[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f2[x],{x,-Infinity,Infinity}]

f3[y_]=Convolve[f[x],f2[x],x,y] //Expand //Together

Integrate[ x f3[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f3[x],{x,-Infinity,Infinity}]

f4[y_]=Convolve[f[x],f3[x],x,y] //Expand

Integrate[ x f4[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f4[x],{x,-Infinity,Infinity}]

Problem #4:

Clear["Global`*"]

f[x_]= (6/10)DiracDelta[x-1]+(4/10)DiracDelta[x+1]

x f[x] //Expand

Integrate[ x f[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f[x],{x,-Infinity,Infinity}]

f2[y_]=Convolve[f[x],f[x],x,y] //Expand

Integrate[ x f2[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f2[x],{x,-Infinity,Infinity}]

f3[y_]=Convolve[f[x],f2[x],x,y] //Expand

Integrate[ x f3[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f3[x],{x,-Infinity,Infinity}]

f4[y_]=Convolve[f[x],f3[x],x,y] //Expand

Integrate[ x f4[x],{x,-Infinity,Infinity}]

Integrate[ (x^2) f4[x],{x,-Infinity,Infinity}]