Visual Representation of the Line Integral

Question

I am trying to better understand the visual interpretation of a Line Integral.

We are all told that a classic integral of a function represents the area under that function between two certain points - but I am trying to now extend this this visual to a line integral.

For example, suppose I want to integrate the function $f(x) = 2x^2$ over the line segment $(0,0)$ and $(1,1)$.

Intuitively, without trying to evaluate the line integral, here is my attempts to represent the line integral (Python code):

  import matplotlib.pyplot as plt
import numpy as np

def f1(x):
    return 2*x**2

def f2(x):
    return x**2 + x**2

x = np.linspace(0, 1, 100)

y1 = f1(x)
y2 = f2(x)

x_line = np.array([0, 1])
y_line1 = np.array([0, 1])
y_line2 = np.array([0, 2])

fig, axs = plt.subplots(1, 2, figsize=(12, 6))

# Attempt 1
axs[0].plot(x, y1, label='Function: $f(x) = 2x^2$', color='blue')
axs[0].plot(x_line, y_line1, label='Line Segment', color='red')

y_line_interp1 = np.interp(x, x_line, y_line1)

# Shade the area between the blue curve and the red line
axs[0].fill_between(x, y1, y_line_interp1, where=(y1 >= y_line_interp1), color='green', alpha=0.3)
axs[0].fill_between(x, y1, y_line_interp1, where=(y1 <= y_line_interp1), color='green', alpha=0.3, label='Line Integral')

axs[0].set_xlabel('x')
axs[0].set_ylabel('y')
axs[0].set_title('Attempt 1: Plot of $f(x) = 2x^2$ and Line Integral')
axs[0].legend()

# Attempt 2
axs[1].plot(x, y2, label='Function: $f(x) = x^2 + x^2$', color='blue')
axs[1].plot(x_line, y_line2, label='Line Segment', color='red')

# Shade the area under the curve
axs[1].fill_between(x, y2, color='green', alpha=0.3, label='Standard Integral')

axs[1].set_xlabel('x')
axs[1].set_ylabel('y')
axs[1].set_title('Attempt 2: Plot of $f(x) = 2x^2$ and Standard Integral')
axs[1].legend()

plt.tight_layout()
plt.show()

My Question: I think Attempt 1 corresponds to the Line Integral whereas Attempt 2 corresponds to the Classic Integral. Have I correctly identified the area corresponding to the Line Integral?

Thanks!

UPDATE: As per the comments from @Anton Vrdoljak, the titles on both graphs have now been changed. As such, they should now be correct (?)