ЛР2

Упражнение 1

import sympy as sp from sympy import * from matplotlib import style from sympy.abc import x style.use('ggplot') y=x*cos(x) plot(y,(x,0,pi/2),adaptive=False, nb_of_points=500)

<sympy.plotting.plot.Plot at 0x3e55c38>

from sympy import * x = Symbol('x') I = integrate(x*cos(x),(x,0,pi/2)) print(I)

-1 + pi/2

Упражнение 2

import sympy as sp from sympy import * from matplotlib import style from sympy.abc import x style.use('ggplot') y=x**2+2*x y1=7-4*x-x**2 plot((y,(x,-5,5)),(y1,(x,-5,5)),adaptive=False, nb_of_points=500)

<sympy.plotting.plot.Plot at 0x50b5c38>

yy=lambda x: x**2+2*x-(7-4*x-x**2) f1=fsolve(yy,-4) f2=fsolve(yy,1) print(f1) print(f2)

[-3.89791576] [0.89791576]

from sympy import * x = Symbol('x') I1 = integrate(x**2+2*x-(7-4*x-x**2),(x,0.89791576,-3.89791576)) print(I1)

36.7680416787309

Пример 1

from numpy import * import matplotlib.pyplot as plt t=linspace(0,2*pi,100) x=cos(t)+1 y=sin(t) plt.plot(x,y) plt.grid(True) plt.show()

Упражнение 3

from numpy import * import matplotlib.pyplot as plt t=linspace(0,2*pi,100) x=cos(t) y=sin(t) xx=2*cos(t) yy=3*sin(t) plt.plot(x,y) plt.plot(xx,yy) plt.grid(True) plt.show()

from sympy import * import numpy as np import sympy as sp t = Symbol('t') I2 = integrate(sin(t)*sp.diff(cos(t)),(t,0,2*pi)) print(np.abs(I2))

pi

from sympy import * t = Symbol('t') I3 = integrate(2*sin(t)*sp.diff(3*cos(t)),(t,0,2*pi)) print(np.abs(I3))

6*pi

S=I3-I2 print(np.abs(S))

5*pi

Пример 2

from numpy import * import matplotlib.pyplot as plt t=linspace(0,2*pi,100) r=t x=r*cos(t)+1 y=r*sin(t) plt.plot(x,y) plt.grid(True) plt.show()

Упражнение 4

from numpy import * import matplotlib.pyplot as plt t=linspace(0,2*pi,100) r=np.e**t x=r*cos(t) y=r*sin(t) plt.plot(x,y) plt.grid(True) plt.show()

t = Symbol('t') Ir=integrate(e**(2*t),(t,0,2*pi)) s=Ir/2 print(s)

71687.5782841632

Упражнение 5

from sympy import * import sympy as sp x=Symbol('x') Ip=integrate(sqrt(1+(sp.diff(x**2))**2),(x,1,2)) print(round(Ip,4))

3.1678

Упражнение 6

from sympy import * t = Symbol('t') Iz=integrate(sqrt(((sp.diff(2*cos(t)))**2)+((sp.diff(3*sin(t)))**2)),(t,0,2*pi)) print(round(Iz,4))

15.8654

Упражнение 7

t = Symbol('t') Il=integrate(sqrt((4*(1+cos(t)))**2+(sp.diff(4*(1+cos(t))))**2),(t,0,2*pi)) print(Il)

4*Integral(sqrt(sin(t)**2 + cos(t)**2 + 2*cos(t) + 1), (t, 0, 2*pi))

Упражнение 8

x=Symbol('x') Ia=integrate((sin(x))**2,(x,0,pi)) IA=Ia*pi print(IA)

pi**2/2

x=Symbol('x') Ib=integrate(x*sin(x),(x,0,pi)) IB=Ib*2*pi print(IB)

2*pi**2

Индивидуальное задание 1

from numpy import * import matplotlib.pyplot as plt t=linspace(0,2*pi,100) x=2*(cos(t))**3 y=2*(sin(t))**3 plt.plot(x,y) plt.grid(True) plt.show()

from sympy import * t = Symbol('t') Ii = integrate(sqrt((sp.diff((2*cos(t))**3))**2+sp.diff((2*sin(t))**3)**2),(t,-2,2)) print(2*Ii)

-48*sqrt(cos(2)**2 + sin(2)**2)*sin(2)**2 + 48

Индивидуальное задание 2

import sympy as sp from sympy import * from matplotlib import style from sympy.abc import x style.use('ggplot') y=atan(x) y1=x-1 plot((y,(x,-5,5)),(y1,(x,-5,5)),adaptive=False, nb_of_points=500, markers=[{'args':[[-5,5],[0,0]],'color':'green','ls':'-'}])

<sympy.plotting.plot.Plot at 0x6b6b0c8>

from sympy import * x = Symbol('x') I = integrate(atan(x)-(x-1),(x,0,2.125)) print(np.abs(I))

1.41667171916890

Упражнение С1

import sympy as sp from sympy import * from matplotlib import style from sympy.abc import x style.use('ggplot') y=x**2+1 y1=3-x plot((y,(x,-5,5)),(y1,(x,-5,5)),adaptive=False, nb_of_points=500, markers=[{'args':[[-5,5],[1,1]],'color':'green','ls':'-'}])

<sympy.plotting.plot.Plot at 0x5eb7cd8>

import sympy as sp from sympy import * import numpy as np from matplotlib import style from sympy.abc import x from scipy.optimize import fsolve Y=lambda x: x**2+1-(3-x) f1=fsolve(Y,-2) f2=fsolve(Y,1) print(f1) print(f2)

[-2.] [1.]

from sympy import * x = Symbol('x') I = integrate(x**2+1-(3-x),(x,-2,1)) print(np.abs(I))

9/2

Упражненеие С2

from numpy import * import matplotlib.pyplot as plt t=linspace(0,2*pi,100) x=(cos(t))**3 y=(sin(t))**3 plt.plot(x,y) plt.grid(True) plt.show()

from sympy import * t = Symbol('t') Ii = integrate(sqrt((sp.diff((cos(t))**3))**2+sp.diff((sin(t))**3)**2),(t,-2,2)) print(2*Ii)

-6*sqrt(cos(2)**2 + sin(2)**2)*sin(2)**2 + 6

Упражнение С3

from numpy import * t=linspace(pi/4,3*pi/4,100) r1=1 r2=2 x1=r1*cos(t) y1=r1*sin(t) x2=r2*cos(t) y2=r2*sin(t) plt.plot(x1,y1,'y') plt.plot(x2,y2,'r') plt.grid() plt.show()

I1=integrate(x-x+2**2,(x,pi/4,3*pi/4)) I2=integrate(x-x+1**2,(x,pi/4,3*pi/4)) S=1/2*(I1-I2) print(np.abs(S))

0.75*pi

Упражнение С4

from sympy import * t = Symbol('t') Iz=integrate(sqrt(((sp.diff(cos(t)))**3)+((sp.diff(sin(t)))**3)),(t,0,2*pi)) print(Iz)

Integral(sqrt(-sin(t)**3 + cos(t)**3), (t, 0, 2*pi))