lijnintegralen

1132 days ago by Marnix.VanDaele

# parametervoorstelling van x^2+y^2=R^2 # m.a.w. we beschrijven een cirkel met straal R en middelpunt (0,0) R=3 t=var('t') p=parametric_plot( (R*cos(t),R*sin(t)) ,(t,0,2*pi),color='red') p.show() 
       
# parametervoorstelling van (x/a)^2+(y/b)^2=1 # m.a.w. we beschrijven een ellips met halve assen van lengte a en b en middelpunt (0,0) a=3 b=2 p=parametric_plot( (a*cos(t),b*sin(t)) ,(t,0,2*pi),color='red') p.show() # deze integraal valt exact te berekenen, vandaar een numerieke berekening numerical_integral(sqrt(a^2*sin(x)^2+b^2*cos(x)^2),0,2*pi) 
        
(15.86543958929059, 1.073539482855681e-08)
(15.86543958929059, 1.073539482855681e-08)
# We bestuderen de doorsnede van een oneindige cylinder (x^2+y^2=R^2) met het vlak cos(alpha)*x+sin(alpha)*(z-z0)=0. # Dit vlak staat horizontaal (en dus dwars op de cylinder als alpha=0 reset() from sage.plot.plot3d.shapes import Cylinder t, y = var('t, y') z0=4 R=5; alpha=2*pi/6 p=Cylinder(R,10, closed=False,color='blue') p+=parametric_plot3d( (R*cos(t),R*sin(t),z0-cos(alpha)/sin(alpha)*R*cos(t)) ,(t,0,2*pi),color='red') p+=plot3d(z0-cos(alpha)/sin(alpha)*x,(x,-R-1,R+1),(y,-R-1,+R+1)) show(p) 
       
# de oppervlakte langs de cirkel x^2+y^2=R^2 onder de functie z=z0-cotg(alpha)*x, kan beschouwd worden als die van een rechthoek # met basis 2*pi*R en hoogte z0 plot(z0-cos(alpha)/sin(alpha)*R*cos(t),(t,0,2*pi))+plot(z0,(t,0,2*pi),linestyle='dashed') R=var('R') z0=var('z0') integrate(R*(z0-cos(alpha)/sin(alpha)*R*cos(t)),(t,0,2*pi)) 
        
2*pi*R*z0
2*pi*R*z0
# hieronder de formule voor de berekening van de lengte van de ellips die de doorsnede vormt # de integraal is echter wederom niet exact te berekenen integrate(sqrt(R^2+(cos(alpha)/sin(alpha)*R*cos(t))^2),(t,0,2*pi)) 
        
#0:
intfudu(exp=%e^-(%i*_SAGE_VAR_t)*sqrt((14*%e^-(2*%i*_SAGE_VAR_t)+%e^-(4*\
%i*_SAGE_VAR_t)+1)*%e^(4*%i*_SAGE_VAR_t)...,%voi=_SAGE_VAR_t)
#1:
extra_integrate(q=%e^-(%i*_SAGE_VAR_t)*sqrt((14*%e^-(2*%i*_SAGE_VAR_t)+%\
e^-(4*%i*_SAGE_VAR_t)+1)*%e^(4*%i*_SAGE_VAR_t)...,x=_SAGE_VAR_t)
Traceback (click to the left of this block for traceback)
...
RuntimeError: ECL says: Error executing code in Maxima:
#0: intfudu(exp=%e^-(%i*_SAGE_VAR_t)*sqrt((14*%e^-(2*%i*_SAGE_VAR_t)+%e^-(4*%i*_SAGE_VAR_t)+1)*%e^(4*%i*_SAGE_VAR_t)...,%voi=_SAGE_VAR_t)
#1: extra_integrate(q=%e^-(%i*_SAGE_VAR_t)*sqrt((14*%e^-(2*%i*_SAGE_VAR_t)+%e^-(4*%i*_SAGE_VAR_t)+1)*%e^(4*%i*_SAGE_VAR_t)...,x=_SAGE_VAR_t)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_19.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("IyBoaWVyb25kZXIgZGUgZm9ybXVsZSB2b29yIGRlIGJlcmVrZW5pbmcgdmFuIGRlIGxlbmd0ZSB2YW4gZGUgZWxsaXBzIGRpZSBkZSBkb29yc25lZGUgdm9ybXQKIyBkZSBpbnRlZ3JhYWwgaXMgZWNodGVyIHdlZGVyb20gbmlldCBleGFjdCB0ZSBiZXJla2VuZW4KaW50ZWdyYXRlKHNxcnQoUl4yKyhjb3MoYWxwaGEpL3NpbihhbHBoYSkqUipjb3ModCkpXjIpLCh0LDAsMipwaSkp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/var/sage/tmpRcVcLf/___code___.py", line 4, in <module>
    exec compile(u'integrate(sqrt(R**_sage_const_2 +(cos(alpha)/sin(alpha)*R*cos(t))**_sage_const_2 ),(t,_sage_const_0 ,_sage_const_2 *pi))
  File "", line 1, in <module>
    
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/misc/functional.py", line 662, in integral
    return x.integral(*args, **kwds)
  File "sage/symbolic/expression.pyx", line 11616, in sage.symbolic.expression.Expression.integral (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/expression.cpp:60290)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 775, in integrate
    return definite_integral(expression, v, a, b, hold=hold)
  File "sage/symbolic/function.pyx", line 996, in sage.symbolic.function.BuiltinFunction.__call__ (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/function.cpp:10984)
  File "sage/symbolic/function.pyx", line 486, in sage.symbolic.function.Function.__call__ (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/function.cpp:6172)
  File "sage/symbolic/function.pyx", line 1085, in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/function.cpp:12234)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 178, in _eval_
    return integrator(*args)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/integration/external.py", line 24, in maxima_integrator
    result = maxima.sr_integral(expression, v, a, b)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py", line 799, in sr_integral
    return max_to_sr(maxima_eval(([max_integrate],[sr_to_max(SR(a)) for a in args])))
  File "sage/libs/ecl.pyx", line 796, in sage.libs.ecl.EclObject.__call__ (/opt/sage/sage-7.4/src/build/cythonized/sage/libs/ecl.c:7712)
  File "sage/libs/ecl.pyx", line 369, in sage.libs.ecl.ecl_safe_apply (/opt/sage/sage-7.4/src/build/cythonized/sage/libs/ecl.c:5336)
RuntimeError: ECL says: Error executing code in Maxima: