题目给出三个已知函数方程：

需要，首先画出函数图像，并且给所围图形打上阴影，并用计算其积分，运行效果如下：

具体代码：

import numpy as np # 给numpy库起个别名
from matplotlib import pyplot as plt  #matplotlib是Python语言及其数值计算库NumPy的绘图库。#设置图像细节
#plt.rcParams['font.sans-serif'] = ['SimHei']  # 用来正常显示中文标签
plt.rcParams['axes.unicode_minus'] = False  # 用来正常显示负号
#plt.legend(loc= 'lower left') # 设置图例位置location
plt.ylim(-10, 10)  # y轴显示区间
plt.xlim(-1, 3)  # x轴显示区间ax = plt.gca() #可以使用 plt.gcf()和 plt.gca()获得当前的图表和坐标轴(分别表示 Get Current Figure 和 Get Current Axes)
ax.xaxis.set_ticks_position('bottom') #设置x轴为下边框
ax.spines['bottom'].set_position(('data', 0))  # spines是脊柱的意思，移动x轴
ax.spines['top'].set_color('none')  # 设置顶部支柱的颜色为空
ax.spines['right'].set_color('none')  # 设置右边支柱的颜色为空
plt.grid(True, linestyle='--', alpha=0.5)  # 网格X1 = np.arange(-1, 3, 0.1)  # x范围
X2 = np.ma.masked_array(X1, mask=((X1 < 0.000001) & (X1 > -0.000001)))  # 反比例函数去掉X为零的情况F1 = np.exp(-X1) + 3  # 指数函数
F2 = 2 * X1 - 2  # 一次函数
F3 = 1 / X2  # 反比例函数def f1(x):  # 指数函数return np.exp(-x) + 3def f2(x):  # 一次函数return 2 * float(x) - 2 # 强制类型转换floatdef f3(x):  # 反比例函数return 1 / float(x)def root(f, g, a, b, eps1):while b - a > eps1:# 二分法 求f = g 方程的根x = (a + b) / 2y = f(x) - g(x)ya = f(a) - g(a)yb = f(b) - g(b)if y * ya < 0:b = xelif y * yb < 0:a = xr = "%.4f" % ((a + b) / 2)return rdef integral(f, a, b, eps2):# 设置初始积分n = 20# n对应的积分值R = [] # 一个空的listI = 0h = (b - a) / n# 计算积分for i in range(0, n - 1):I += (f(a + (i + 0.5) * h))R.append(h * I) # 记录h*I 的值while (1):# 更新n *= 2I = 0h = (b - a) / n# 计算for i in range(0, n - 1): # 从0 ～n-1的循环I += (f(a + (i + 0.5) * h)) # 矩形公式R.append(h * I)# 判断if (abs(R[-1] - R[-2]) / 3 < eps2):breakc = []  # 存下每个积分点的横坐标for i in range(0, n - 1):c.append(a + h * i)# print("R:", R[-1])return c, R[-1]def drawRomberg(f1, f2, a, b, eps2, n): #画出求积部分x1, y1 = integral(f1, a, b, eps2)x1_ = []for i in range(len(x1)):if i % n == 0:x1_.append(x1[i])if i == len(x1) - 1:x1_.append(x1[i])for i in x1_:line = [i, f1(i), i, f2(i)]plt.plot(line[::2], line[1::2], color='red',linewidth=1, linestyle="--")if __name__ == '__main__': #“if __name__==’__main__:”也像是一个标志，象征着Java等语言中的程序主入口，告诉其他程序员，代码入口在此——这是“if __name__==’__main__:”这条代码的意义之一。# 计算四个交点的坐标x4 = "%.4f" % float(root(f1, f2, 2, 3, 0.001))y4 = "%.4f" % float(f2(x4))x2 = "%.4f" % float(root(f1, f3, 0.1, 1, 0.001))y2 = "%.4f" % float(f3(x2))x3 = "%.4f" % float(root(f2, f3, 1, 2, 0.001))y3 = "%.4f" % float(f2(x3))x1 = "%.4f" % float(root(f2, f3, -1, -0.1, 0.001))y1 = "%.4f" % float(f2(x1))print("x1:%.4f" % float(x1), "y1:%.4f" % float(y1))print("x2:%.4f" % float(x2), "y2:%.4f" % float(y2))print("x3:%.4f" % float(x3), "y3:%.4f" % float(y3))print("x4:%.4f" % float(x4), "y4:%.4f" % float(y4))# 画出初始的三条函数plt.plot(X1, F1, linewidth=1.5, linestyle="-", label="f1")plt.plot(X1, F2, linewidth=1.5, linestyle="-", label="f2")plt.plot(X2, F3, linewidth=1.5, linestyle="-", label="f3")# 画出四个交点围成的三角形polygon = [[float(x2), float(y2)], [float(x3), float(y3)], [float(x4), float(y4)]]# line_area = polygon+[polygon[0]]# plt.plot([i[0] for i in line_area], [i[1] for i in line_area])# 画出三个焦点plt.scatter(polygon[0][0], polygon[0][1], s=50, c='r')plt.scatter(polygon[1][0], polygon[1][1], s=50, c='r')plt.scatter(polygon[2][0], polygon[2][1], s=50, c='r')# 计算面积I = integral(f1, float(x2), float(x4), 0.001)[1] - integral(f2, float(x3),float(x4), 0.001)[1] - \integral(f3, float(x2), float(x3), 0.001)[1]# 画出求积部分drawRomberg(f1, f3, float(x2), float(x3), 0.001, 80)drawRomberg(f1, f2, float(x3), float(x4), 0.001, 80)# 显示计算结果plt.text(1.5, 6, "I = {}".format("%.4f" % float(I)))# 显示plt.legend(loc='upper left')plt.show()