第四篇：不定积分与定积分

不定积分

不定积分的性质

基本积分表

积分公式：https://baike.baidu.com/item/%E7%A7%AF%E5%88%86%E5%85%AC%E5%BC%8F/8556651?fr=aladdin

from sympy import integrate,diff,log,cos,Rational
from sympy.abc import x,u
print(integrate(1/(x*x**Rational(1,3)),x))
print(integrate(x**u,x))

定积分

定积分定义

定积分计算方法（牛顿 – 莱布尼茨公式）

from sympy import integrate,diff,log,cos,Rational,cos,sin,pi
from sympy.abc import x,u
print(\'例一：{}\'.format(integrate(1/x,(x,-2,-1))))
print(\'例二：{}\'.format(integrate(2*cos(x)+sin(x)-1,(x,0,pi/2))))

import scipy.integrate as sci
import numpy as np
import matplotlib.pyplot as plt
import matplotlib
from matplotlib.patches import Polygon

myfont = matplotlib.font_manager.FontProperties(fname=r\'C:\Windows\Fonts\SimHei.ttf\')  # 显示中文的设置[3]


def f(x):
    return 2*np.cos(x) + np.sin(x) - 1


a = 0
b = np.pi/2
# linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None)    #np.linspace函数的参数,默认为分为50段
x = np.linspace(-1,5)
y = f(x)

fig, ax = plt.subplots(figsize=(7, 5))
plt.plot(x, y, \'b\', linewidth=2,label=\'f(x) = sin(x) + 0.5*x\')
plt.ylim(ymin=0)

Ix = np.linspace(a, b)
Iy = f(Ix)
verts = [(a, 0)] + list(zip(Ix, Iy)) + [(b, 0)]
poly = Polygon(verts, facecolor=\'0.7\', edgecolor=\'0.5\')  # 绘制曲线阴影部分
ax.add_patch(poly)
# matplotlib.axes._subplots.AxesSubplot

# labels
plt.text(0.5 * (a + b), 0.5, r"$\int_a^b f(x)dx$", horizontalalignment=\'center\', fontsize=20)
plt.figtext(0.9, 0.075, \'$x$\')
plt.figtext(0.075, 0.9, \'$f(x)$\')

plt.legend(loc=\'upper left\')
ax.set_xticks((a, b))
ax.set_xticklabels([\'$a$\', \'$b$\'])
ax.set_yticks([f(a), f(b)])
plt.title(\'积分图\', fontproperties=myfont)
plt.show()

二重积分的概念

二重积分的计算

选择积分次序的原则

(1)积分容易。

(2)尽量少分块或不分块。

from sympy import integrate
from sympy.abc import x,y
f = x*y
iy = integrate(f,(y,1,x))
ix = integrate(iy,(x,1,2))
print(ix)

 

解法2：先对y积分则需要分块