% Computes the integral of arctan(x) from 0 to 1 which
% equals pi/4-1/2*ln(2) 
clc
format compact
format long
a=0, b=1, n1=10
dx1=(b-a)/n1
x1=[a+dx1:dx1:b] % Break up the interval into 10 parts
y1=atan(x1) % 10 rectangles based upon height of right endpoint
ymid1=atan(x1-dx1/2) % 10 rectangles using the midpoint as your height
sum(y1)*dx1 % sum of right rectangles
sum(ymid1)*dx1 % sum of midpoint rectangles
% Repeat with 1000 rectangles
n2=1000
dx2=(b-a)/n2
x2=[a+dx2:dx2:b]; % Note the semicolon
y2=atan(x2); % Another semicolon
ymid2=atan(x2-dx2/2);
sum(y2)*dx2
sum(ymid2)*dx2
exact=pi/4-1/2*log(2)
% How do these values compare with the exact value?