Solution for Problem 6

Input := 

Clear[y, i]

Input := 

dx = 0.5;
n = 20;
y[i_] = height[i/2];

Input := 

TrapRule20 =
	N[(dx/2) (y[0] + 2 Sum[y[i], {i, 1, n-1}] + y[n])]

Output =

58.2025

Input := 

SimpRule20 =
	N[(dx/3)
		(4 Sum[y[2 i - 1], {i, 1, n/2}] +
		 2 Sum[y[2 i], {i, 1, n/2 - 1}])]

Output =

58.3046

Here's another shot to see if higher n can improve the estimates more.

Input := 

Clear[y, i]

Input := 

dx = 0.25;
n = 40;
y[i_] = height[i/4];

Input := 

TrapRule40 =
	N[(dx/2) (y[0] + 2 Sum[y[i], {i, 1, n-1}] + y[n])]

Output =

58.293

Input := 

SimpRule40 =
	N[(dx/3)
		(4 Sum[y[2 i - 1], {i, 1, n/2}] +
		 2 Sum[y[2 i], {i, 1, n/2 - 1}])]

Output =

58.3232