检查y=6x与y=6x²围成区域及绕轴旋转绘图正确性并添加图例

问题说明 #

编写Python代码绘制$y=6x$与$y=6x^{2$围成的有界区域，并生成该区域绕x轴、y轴旋转形成的旋转体。需要检查代码绘图正确性，添加图例明确蓝色线条对应$y=6x$，橙色线条对应$y=6x}2$。

原代码 #

# Compare the plot at xy axis with the solid of revolution toward x and y axis
# For region bounded by the line y = 6x and y = 6x^2
import matplotlib.pyplot as plt
import numpy as np

n = 100

fig = plt.figure(figsize=(14, 7))
ax1 = fig.add_subplot(221)
ax2 = fig.add_subplot(222, projection='3d')
ax3 = fig.add_subplot(223)
ax4 = fig.add_subplot(224, projection='3d')

y = np.linspace(0, 10, n)
x1 = (y/6) 
x2 = (y/6) ** (1/2)
t = np.linspace(0, np.pi * 2, n)

u = np.linspace(-1, 2, 60)
v = np.linspace(0, 2*np.pi, 60)
U, V = np.meshgrid(u, v)

X = U
Y1 = (6*U**2)*np.cos(V)
Z1 = (6*U**2)*np.sin(V)

Y2 = (6*U)*np.cos(V)
Z2 = (6*U)*np.sin(V)

Y3 = ((1/6)*U**(1/2))*np.cos(V)
Z3 = ((1/6)*U**(1/2))*np.sin(V)

Y4 = (U/6)*np.cos(V)
Z4 = (U/6)*np.sin(V)

xn = np.outer(x1, np.cos(t))
yn = np.outer(x1, np.sin(t))
zn = np.zeros_like(xn)
for i in range(len(x1)):
    zn[i:i + 1, :] = np.full_like(zn[0, :], y[i])

ax1.plot(x1, y, x2, y)
ax1.set_title("$f(x)$")
ax2.plot_surface(X, Y3, Z3, alpha=0.3, color='red', rstride=6, cstride=12)
ax2.plot_surface(X, Y4, Z4, alpha=0.3, color='blue', rstride=6, cstride=12)
ax2.set_title("$f(x)$: Revolution around $y$")

# find the inverse of the function
x_inverse = y
y1_inverse = np.power(x_inverse, 1)
y2_inverse = np.power(x_inverse, 1 / 2)

ax3.plot(x_inverse, y1_inverse, x_inverse, y2_inverse)
ax3.set_title("Inverse of $f(x)$")


ax4.plot_surface(X, Y1, Z1, alpha=0.3, color='red', rstride=6, cstride=12)
ax4.plot_surface(X, Y2, Z2, alpha=0.3, color='blue', rstride=6, cstride=12)
ax4.set_title("$f(x)$: Revolution around $x$")

plt.tight_layout()
plt.show()

原代码问题 #

• 坐标范围错误：$y=6x$与$y=6x^2$交点为$(0,0)$和$(1,6)$，原代码绘制了超出该范围的无效区域
• 旋转体计算错误：绕y轴旋转的变量对应关系混乱，绕x轴旋转的x范围超出有界区域
• 图例缺失：2D图无图例，无法区分两条曲线对应的函数
• 逆函数绘图错误：ax3未正确绘制原函数的逆函数曲线

修正后的代码 #

import matplotlib.pyplot as plt
import numpy as np

n = 100

fig = plt.figure(figsize=(14, 7))
ax1 = fig.add_subplot(221)
ax2 = fig.add_subplot(222, projection='3d')
ax3 = fig.add_subplot(223)
ax4 = fig.add_subplot(224, projection='3d')

# -------------------------- 2D有界区域（x关于y的函数，绕y轴旋转基础） --------------------------
y = np.linspace(0, 6, n)
x_line = y / 6             # y=6x的逆函数：x=y/6
x_parabola = np.sqrt(y / 6)# y=6x²的逆函数：x=√(y/6)

# 绘制曲线并添加图例
line_plot, = ax1.plot(x_line, y, color='blue', label='$y=6x$')
parabola_plot, = ax1.plot(x_parabola, y, color='orange', label='$y=6x^2$')
ax1.legend(handles=[line_plot, parabola_plot])
ax1.fill_betweenx(y, x_line, x_parabola, alpha=0.1, color='gray')
ax1.set_title('有界区域（x-y轴）')
ax1.set_xlabel('x')
ax1.set_ylabel('y')

# -------------------------- 绕y轴旋转的旋转体 --------------------------
u_y = np.linspace(0, 6, 60)
v = np.linspace(0, 2*np.pi, 60)
U_y, V = np.meshgrid(u_y, v)

# y=6x绕y轴旋转
X_line_y = (U_y / 6) * np.cos(V)
Z_line_y = (U_y / 6) * np.sin(V)
Y_line_y = U_y

# y=6x²绕y轴旋转
X_parabola_y = np.sqrt(U_y / 6) * np.cos(V)
Z_parabola_y = np.sqrt(U_y / 6) * np.sin(V)
Y_parabola_y = U_y

surf_line_y = ax2.plot_surface(X_line_y, Y_line_y, Z_line_y, alpha=0.3, color='blue', rstride=6, cstride=12)
surf_parabola_y = ax2.plot_surface(X_parabola_y, Y_parabola_y, Z_parabola_y, alpha=0.3, color='orange', rstride=6, cstride=12)
ax2.legend([surf_line_y, surf_parabola_y], ['$y=6x$ 绕y轴旋转', '$y=6x^2$ 绕y轴旋转'], loc='upper right')
ax2.set_title('绕y轴旋转的旋转体')
ax2.set_xlabel('x')
ax2.set_ylabel('y')
ax2.set_zlabel('z')

# -------------------------- 原函数2D绘图（y关于x的函数，绕x轴旋转基础） --------------------------
x = np.linspace(0, 1, n)
y_line = 6 * x            # y=6x
y_parabola = 6 * x**2     # y=6x²

line_plot_inv, = ax3.plot(x, y_line, color='blue', label='$y=6x$')
parabola_plot_inv, = ax3.plot(x, y_parabola, color='orange', label='$y=6x^2$')
ax3.legend(handles=[line_plot_inv, parabola_plot_inv])
ax3.fill_between(x, y_line, y_parabola, alpha=0.1, color='gray')
ax3.set_title('原函数曲线（x-y轴）')
ax3.set_xlabel('x')
ax3.set_ylabel('y')

# -------------------------- 绕x轴旋转的旋转体 --------------------------
u_x = np.linspace(0, 1, 60)
U_x, V = np.meshgrid(u_x, v)

# y=6x绕x轴旋转
Y_line_x = 6 * U_x * np.cos(V)
Z_line_x = 6 * U_x * np.sin(V)
X_line_x = U_x

# y=6x²绕x轴旋转
Y_parabola_x = 6 * U_x**2 * np.cos(V)
Z_parabola_x = 6 * U_x**2 * np.sin(V)
X_parabola_x = U_x

surf_line_x = ax4.plot_surface(X_line_x, Y_line_x, Z_line_x, alpha=0.3, color='blue', rstride=6, cstride=12)
surf_parabola_x = ax4.plot_surface(X_parabola_x, Y_parabola_x, Z_parabola_x, alpha=0.3, color='orange', rstride=6, cstride=12)
ax4.legend([surf_line_x, surf_parabola_x], ['$y=6x$ 绕x轴旋转', '$y=6x^2$ 绕x轴旋转'], loc='upper right')
ax4.set_title('绕x轴旋转的旋转体')
ax4.set_xlabel('x')
ax4.set_ylabel('y')
ax4.set_zlabel('z')

plt.tight_layout()
plt.show()

修正要点 #

• 范围调整：将x限制在$[0,1]$、y限制在$[0,6]$，仅绘制两条曲线围成的有效区域
• 图例完善：2D和3D图均添加图例，明确蓝色对应$y=6x$，橙色对应$y=6x^2$
• 旋转体修正：
  • 绕x轴旋转：以x为自变量，将y值绕x轴生成旋转曲面
  • 绕y轴旋转：以y为自变量，将x值绕y轴生成旋转曲面
• 视觉优化：填充有界区域增加透明度，提升图形直观性

内容的提问来源于stack exchange，提问作者Freya the Goddess