4449.sagews

# integrate z^2 on the unit circle

f(z) = z^2 
# define the integrand

c(t)=e^(I*t) 
# parametrize the unit circle. c(t):[0,2pi]

start=0
stop=2*float(pi)
N=1000

line_points = srange(float(start), float(stop), (stop-start)/N,include_endpoint=True) 
#N+1 points on [start, stop] 
z = map(c,line_points) 
#N+1 points on curve 
#The points are denoted z[0], z[1], ..., z[N]

sum(f(z[i])*(z[i+1]-z[i]) for i in range(0,N-1)) # approximate integral of z^2

-9.86918230106e-05 - 0.00628239983217*I

dc(t)=c.diff(t) # derivative of c(t)
answer=integrate(f(c(t))*dc(t),(t,start,stop))
     # integrate using parametrization 
answer

-2.44929359829e-16*I

float(answer.real_part())+float(answer.imag_part())*I 
    # check if answer from parametrization agrees with approximation by taking its floating-point value

-2.44929359829e-16*I