Calculus, Chapter 16.4 problem 16

16.4.16.py

# -*- coding: utf-8 -*-
# Warning: takes FOREVER to run!
from sympy.core import Symbol, diff, pi, sympify
from sympy.integrals import integrate
from sympy.functions import sin, cos, sqrt

x, y, t = map(Symbol, 'xyt')
P = 2*x - (x**3) * (x**5)
Q = (x**3) * (y ** 8)

# Line integral

r = (cos(t), 2 * sin(t))
integrand = (P.subs({x: r[0], y: r[1]}) * diff(r[0], t) +
             Q.subs({x: r[0], y: r[1]}) * diff(r[1], t))

print "LINE INTEGRAL"
print
print "Path:", r
print "Integrand:", integrand
print "Result:", integrate(integrand, (t, 0, 2 * pi))

# Green’s theorem

integrand = diff(Q, x) - diff(P, y)
xlimits = (x, -1, 1)
ylimits = (y, -2 * sqrt(1 - x**2), 2 * sqrt(1 - x**2))
print "GREEN’S THEOREM"
print
print "Integrand:", integrand
print "Limits: x:", xlimits
print "        y:", ylimits
print integrate(integrand, ylimits, xlimits)