= Symbol('t', positive=True)
u_expr = 2 + t + t**2

x = Rational(11, 3) - u_expr
y = 2*u_expr - 4

print(f"x(t) = {x}")
print(f"y(t) = {y}")

--- 2. Compute derivatives ---

xp = diff(x, t) # dx/dt
yp = diff(y, t) # dy/dt

Arc length integrand: sqrt(xp² + yp²)

integrand = sqrt(xp2 + yp2)
integrand_simplified = simplify(integrand)

print(f"\nx'(t) = {xp}")
print(f"y'(t) = {yp}")
print(f"Speed = sqrt(x'² + y'²) = {integrand_simplified}")

--- 3. Integrate from t=0 to t=1 ---

L = integrate(integrand_simplified, (t, 0, 1))
L_exact = simplify(L)
L_numeric = float(L_exact)

print(f"\nArc length L = {L_exact}")
print(f"L ≈ {L_numeric:.6f}")