4436.sagews

#Integrate z on the unit circle from 0 to 2pi

f(z) = z  # define the integrand

c(t) = e^(I*t) # parameterize the 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) 

z = map(c,line_points)

sum(f(z[i])*(z[i+1]-z[i]) for i in range(0,N-1))  # The approximating formula

-5.92166523229e-05 - 0.00628289591791*I

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

# As we can see based on the solutions of the approximating sum, the parameterization, we see that the solution is very very close to "0"