import numpy as np
import matplotlib.pyplot as plt
 
# 参数
k = 5 * np.sqrt(3) / 3   # 由体积相等算出的 k
sqrt3 = np.sqrt(3)
 
x = np.linspace(0, sqrt3, 400)
y_parabola = x**2
y_line = k * x
y_top = 3 * np.ones_like(x)
 
# 交点横坐标
x_inter = 3 / k   # 直线与 y=3 的交点，此时约等于 1.039
x_intersect_parabola = sqrt3   # 抛物线顶点与 y=3 交点
 
plt.figure(figsize=(6, 5))
 
# 填充区域 D (整体)
plt.fill_between(x, y_parabola, y_top, where=(x <= sqrt3), color='lightgray', alpha=0.5, label='$D$ (整体)')
 
# 再分别用不同透明度区分 D1 和 D2
# D2: 在 0 ≤ x ≤ x_inter 上，从直线到 y=3
x2 = x[x <= x_inter]
y2_lower = k * x2
plt.fill_between(x2, y2_lower, 3, color='skyblue', alpha=0.7, label='$D_2$ (绕 x 轴旋转)')
# D1: 分两段：0~x_inter 上从抛物线到直线；x_inter~sqrt3 上从抛物线到 y=3
x1a = x[x <= x_inter]
y1a_lower = x1a**2
y1a_upper = k * x1a
x1b = x[(x >= x_inter) & (x <= sqrt3)]
y1b_lower = x1b**2
plt.fill_between(x1a, y1a_lower, y1a_upper, color='lightcoral', alpha=0.7, label='$D_1$ (绕 x 轴旋转)')
plt.fill_between(x1b, y1b_lower, 3, color='lightcoral', alpha=0.7)
 
# 绘制曲线
plt.plot(x, y_parabola, 'b-', linewidth=2, label='$y = x^2$')
plt.plot(x, y_line, 'r--', linewidth=2, label=f'$y = kx,\ k = {k:.3f}$')
plt.axhline(y=3, color='green', linestyle='--', linewidth=2, label='$y = 3$')
plt.axvline(x=0, color='black', linewidth=0.5)
 
# 标注交点
plt.plot([x_inter, x_inter], [0, 3], 'k:', linewidth=0.8)
plt.plot(x_inter, 3, 'ko', markersize=5)
plt.plot(sqrt3, 3, 'ko', markersize=5)
plt.plot(0, 0, 'ko', markersize=3)
 
plt.xlim(0, sqrt3 + 0.2)
plt.ylim(0, 3.2)
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.title('区域 $D$ 被直线 $y=kx$ 分成 $D_1$ 和 $D_2$')
plt.legend(loc='upper left')
plt.grid(alpha=0.3)
plt.gca().set_aspect('equal', adjustable='box')
plt.show()

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