% Math 551 demo 4_17_12 on trapezoidal rule for integrating 
% y = x^2, y=x^(0.5), and y=x^3 from 0 to 1 using trapz,
% traptd2.m and simtd2.m 


format long
h=.1;
y=(0:h:1).^(1.0);
h*trapz(y)

ans =

   0.500000000000000

y=(0:h:1).^(2.0);
h*trapz(y)

ans =

   0.335000000000000

h=.01;
y=(0:h:1).^(2.0);
h*trapz(y)

ans =

   0.333350000000000

h=.001;
y=(0:h:1).^(2.0);
h*trapz(y)

ans =

   0.333333500000000

h=.0001;
y=(0:h:1).^(2.0);
h*trapz(y)

ans =

   0.333333335000000

clear
h=.1;
y=(0:h:1).^(0.5);
h*trapz(y)

ans =

   0.660509341706817

h=.01;
y=(0:h:1).^(0.5);
h*trapz(y)

ans =

   0.666462947103148

h=.001;
y=(0:h:1).^(0.5);
h*trapz(y)

ans =

   0.666660134393682

h=.0001;
y=(0:h:1).^(0.5);
h*trapz(y)

ans =

   0.666666459197108

y=(0:h:1).^(3.0);
h*trapz(y)

ans =

   0.250000002500000

>> f = @(x) x.^2

f = 

    @(x)x.^2

>> traptd2(f,0,1,11)

ans =

   0.335000000000000

>> simptd2(f,0,1,11)

ans =

   0.333333333333333

>> f = @(x) x.^(0.5)

f = 

    @(x)x.^(0.5)

>> simptd2(f,0,1,11)

ans =

   0.664099589757421

>> f = @(x) x.^3

f = 

    @(x)x.^3

>> traptd2(f,0,1,11)

ans =

   0.252500000000000

>> simptd2(f,0,1,11)

ans =

   0.250000000000000

>> f = @(x) x.^4

f = 

    @(x)x.^4

>> traptd2(f,0,1,11)

ans =

   0.203330000000000

>> simptd2(f,0,1,11)

ans =

   0.200013333333333

>> f = @(x) x.^(0.5)

f = 

    @(x)x.^(0.5)

>> simptd2(f,0,1,101)

ans =

   0.666585482066724

>> simptd2(f,0,1,1001)

ans =

   0.666664099383543

>> simptd2(f,0,1,10001)

ans =

   0.666666585482046


diary off