>>> from sympy import *
>>> from scipy.integrate import quad
>>> t=var('t')
>>> ds=sqrt((2*t)**2+(2*sin(2*t))**2+(2*cos(t))**2)
>>> m=quad(lambdify(t,ds),(t,0,4*pi))[0]
>>> m
161.64342911399328
>>> xm=quad(lambdify(t,t**2*ds),0,4*pi)[0]
>>> xm/m
77.6179451479088
>>> ym=quad(lambdify(t,cos(2*t)*ds),0,4*pi)[0]
>>> ym/m
0.00714883706077915
>>> zm=quad(lambdify(t,2*sin(t)*ds),0,4*pi)[0]
>>> zm/m
-0.300157717268558
>>> gdr=lambda x,y: x*y*diff(x,t)-2*diff(y,t)
>>> x1,x2,x3,x4=3*cos(t),3+5*t,8-5*t,3
>>> y1,y2,y3,y4=3*sin(t),2*t,2+2*t,4*t
>>> integrate(fdr(x1,y1),(t,0,2*pi))
-18*pi
>>> integrate(fdr(x2,y2),(t,0,1))
-6
>>> integrate(fdr(x3,y3),(t,0,1))
-26
>>> integrate(fdr(x4,y4),(t,0,1))
-12
>>> integrate(gdr(x1,y1),(t,0,2*pi))
0
>>> integrate(gdr(x2,y2),(t,0,1))
83/3
>>> integrate(gdr(x3,y3),(t,0,1))
-247/3
>>> 83-247
-164
>>> integrate(gdr(x4,y4),(t,0,1))
-8
>>> N(1/sqrt(2)-1/sqrt(5))
0.259893185686590
>>> x=(2+cos(7*t))*cos(t)
>>> y=(3+cos(7*t))*sin(t)
>>> dx=diff(x,t)
>>> dy=diff(y,t)
>>> da=lambdify(t,x*dy)
>>> xda=lambdify(t,x**2/2*dy)
>>> yda=lambdify(t,-y**2/2*dx)
>>> r2da=lambdify(t,(x**3*dy-y**3*dx)/3)
>>> a=quad(da,0,2*pi)[0]
>>> a
20.420352248333664
>>> N(13/2*pi)
20.4203522483337
>>> xa=quad(xda,0,2*pi)[0]
>>> ya=quad(yda,0,2*pi)[0]
>>> r2a=quad(r2da,0,2*pi)[0]
>>> xa,ya,r2a
(-3.6089187194221495e-14, 2.3648660679138e-14, 91.30253649495339)
>>> r2a/a
4.471153846153846
