7.6 9 19
#9a
> f:=exp(x);                               
                                  f := exp(x)

> exactf:=evalf(int(f,x=0..4));            
                             exactf := 53.59815003

> left2:=evalf(value(leftsum(f,x=0..4,2)));
                             left2 := 16.77811220

> right2:=evalf(value(rightsum(f,x=0..4,2)));
                             right2 := 123.9744123

> trap2:=evalf(value(trapezoid(f,x=0..4,2)));
                             trap2 := 70.37626223

> mid2:=evalf(value(middlesum(f,x=0..4,2))); 
                              mid2 := 45.60763750

> simp2:=evalf(value(simpson(f,x=0..4,4)));
                             simp2 := 53.86384573

#9b
> left4:=evalf(value(leftsum(f,x=0..4,4)));  
                             left4 := 31.19287485

> right4:=evalf(value(rightsum(f,x=0..4,4)));
                             right4 := 84.79102488

> trap4:=evalf(value(trapezoid(f,x=0..4,4)));
                             trap4 := 57.99194987

> mid4:=evalf(value(middlesum(f,x=0..4,4))); 
                              mid4 := 51.42835626

> simp4:=evalf(value(simpson(f,x=0..4,8)));
                             simp4 := 53.61622078


#9c
> leftImprove:=(left2-exactf)/(left4-exactf);
                          leftImprove := 1.643364678

> rightImprove:=(right2-exactf)/(right4-exactf);
                          rightImprove := 2.256164672

> trapImprove:=(trap2-exactf)/(trap4-exactf);   
                          trapImprove := 3.818588195

> midImprove:=(mid2-exactf)/(mid4-exactf);   
                           midImprove := 3.682613823

> simpImprove:=(simp2-exactf)/(simp4-exactf);
                          simpImprove := 14.70308095

Left/Right should be 2, Trap/Mid should be 4, Simpson should be 16
The answers are a little off because of the rapid growth of e^x.

#19a
Increasing have left < mid,trap,true < right
Smiling have mid < true < trap

Looks like increasing and frowning.
        err n=3     err n=30
Left    2.205271    0.2205271
Right   1.686549    0.1686549
Trap    0.259361    0.00259361
Mid     0.121030    0.0012103
Simp    0.005767l   0.0000005767l

7.7 3
#3
int(x/(4+x^2),x=1..Large) => u = x^2+4; du = 2x dx; x=0=>u=4;
x=Large=>u=Large. So 0.5 * int(1/u,u=4..Large) =
limit( ln Large - ln 4 ) -> infinity as Large -> infinity so the int diverges.

#10
int(1/(1+x^2),x=1..Large) = arctan(Large) - arctan(1) => pi/2-pi/4= pi/4 

7.8 5
#5
x^3+1 > x^3 so 1/(x^3+1) < 1/(x^3) and since int(x^(-3),x=1..infinity)
converges so does int(1/(x^3+1),x=1..infinity)