%% Definite and indefinite integrals
% Nick Trefethen, October 2012

%%
% (Chebfun example calc/Integrals.m) 
% [Tags: #integral, #SUM, #CUMSUM]

%%
% Suppose we have a function, like this one:
x = chebfun('x',[0 10]);
f = round(2*cos(x));
plot(f), ylim(2.5*[-1 1])

%%
% The Chebfun command `sum` returns the definite integral over the prescribed
% interval, which is just a number:
format long, sum(f)

%%
% You can also calculate the definite interval over a subinterval by giving
% two additional arguments, like this:
sum(f,3,4)

%%
% To compute an indefinite integral, use the Chebfun command `cumsum`.
% This returns a chebfun defined over the given interval:
g = cumsum(f);
plot(g,'m')

%%
% Thus another way to compute the integral over a subinterval would be to take
% the difference of two values of the cumsum:
g(4) - g(3)

%%
% As always in calculus, when working with indefinite integrals you must be
% careful to remember the arbitrary constant that may be added.  Thus for
% example, if you integrate $f$ and then differentiate it, you get $f$ back
% again:
norm( diff(cumsum(f)) - f )

%%
% If you differentiate $f$ and then integrate it, on the other hand, you get
% something different:
norm( cumsum(diff(f)) - f )

%%
% Plotting the two instantly alerts us that we forgot to add back in the value
% at the left endpoint, namely $f(0) = 2$:
plot(f,'b',cumsum(diff(f)),'r')

%%
% Sure enough, adding this number makes the two functions agree:
norm( f(0)+cumsum(diff(f)) - f)

