Abstract Code Python import micropip await micropip.install("matplotlib") await micropip.install("numpy") import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Define the limits of the rectangle R in the xy-plane x_min, x_max = 0, 2 # Limits for x y_min, y_max = 0, 3 # Limits for y # Create a grid of points in the xy-plane (for R) x = np.linspace(x_min, x_max, 30) y = np.linspace(y_min, y_max, 30) X, Y = np.meshgrid(x, y) # Define a surface z = f(x, y), an arbitrary function for the example Z = np.sin(X) * np.cos(Y) # Plotting the surface and the rectangle fig = plt.figure() ax = fig.add_subplot(111, projection='3d') # Plot the surface S ax.plot_surface(X, Y, Z, cmap='Blues', edgecolor='k', alpha=0.7) # Highlight the rectangle R in the xy-plane (z = 0) ax.plot([x_min, x_max, x_max, x_min, x_min], [y_min, y_min, y_max, y_max, y_min], [0, 0, 0, 0, 0], color='green', lw=3) # Plot vertical lines from the surface to the xy-plane (z = 0) for i in range(0, X.shape[0], 5): # Plot every 5th line for clarity ax.plot([X[i, 0], X[i, 0]], [Y[i, 0], Y[i, 0]], [0, Z[i, 0]], 'k--', lw=1) ax.plot([X[i, -1], X[i, -1]], [Y[i, -1], Y[i, -1]], [0, Z[i, -1]], 'k--', lw=1) # Labels and title ax.set_xlabel('X axis') ax.set_ylabel('Y axis') ax.set_zlabel('Z axis') ax.set_title('Double Integral Visualization Over a Rectangle R') plt.show() Output