Length of a parametric curve (integral of differential of the curve)

Question

I am trying to learn differential geometry and sympy for the first time. Using sympy , I am able to define expressions for parametric curve and find velocities. I am trying to compute the length of the curve as per the below definition, but unable to figure out, how to do this with sympy. Can someone please provide pointers on how to compute the curve length.

Below is the current code I have (using sympy & matplotlib).

# Expression of Parametric Curve 
t = sympy.symbols('t')
x_expr = sympy.cos(t)
y_expr = sympy.sin(t)

f_x = sympy.lambdify(t, x_expr, 'numpy')
f_y = sympy.lambdify(t, y_expr, 'numpy')

# Differential of Parametric Curve 
diff_x = sympy.lambdify(t, sympy.diff(x_expr, t), 'numpy')
diff_y = sympy.lambdify(t, sympy.diff(y_expr, t), 'numpy')

t_values = np.linspace(0, 2 * np.pi, 80, endpoint=True)

X = f_x(t_values)
Y = f_y(t_values)

# Plot the curve
clear_figure()
ax = create_subplot()
ax.set_xlim(-1.5, 1.5)
ax.set_ylim(-1.5, 1.5)
ax.scatter(X, Y, s=4)
plt_canvas.draw()

# Plot the velocities 
t1_values = np.linspace(0, 2 * np.pi, 9, endpoint=True)
for t1 in t1_values:
    ax.arrow(f_x(t1), f_y(t1), diff_x(t1), diff_y(t1), head_width=0.02, head_length=0.02, ec='red', linewidth=0.1)
plt_canvas.draw()

Answer 1

From geometry you can define a parametric curve and it can tell you the length of the same. Is this what you were expecting:

>>> Curve((cos(t), sin(t)), (t, 0, 2*pi)).length
2*pi