Part a

f[x_]=(1/(Sqrt[2π]))(E^(-(x^2)/2));
Plot[f[x],{x,-10,10},PlotRange->All]

Part b

a=1;b=3;n=8;
P=Table[k,{k,a,b,((b-a)/n)}];
N[(((b-a)/n))(∑_(i=1)^n)((f[P[[i+1]]]+f[P[[i]]])/2)]

Part c

N[(((b-a)/(3n)))(f[a]+(∑_(i=1)^(n/2-1))2f[P[[2 i]]]+(∑_(i=2)^(n/2))4f[P[[2i-1]]]+f[b])]

Part d

This is based on the error bound |E|≤((((b-a))^3)/(12(n^3)))MAX[|(f^((2)))(x)|], meaning:
(n^3)>((((b-a))^3)/(12|E|))MAX[|(f^((2)))(x)|

Solve[f'''[x]==0]//N

e=0.00005;
p=1.7320508075688772;
f2=Max[Abs[{f''[1],f''[3],f''[p]}]];
Ceiling[((((((b-a))^3)/(12e))f2)^(1/3))]

Part e

This is based on the error bound |E|≤((((b-a))^5)/(180(n^4)))MAX[|(f^((4)))(x)|], meaning:
(n^4)>((((b-a))^5)/(180|E|))MAX[|(f^((4)))(x)|

Solve[f'''''[x]==0]//N

e=0.00005;
p=1.3556261799742657;
f4=Max[Abs[{f''''[1],f''''[3],f''''[p]}]];
Ceiling[((((((b-a))^5)/(180 e))f4)^(1/4))]

Part f

NIntegrate[f[x],{x,1,3}]