To calculate the length of the curve $y = x^{1/3}$ from $(0, 1)$ to $(1, 0)$, we use the arc length formula:

$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

Here:

• $y = x^{1/3}$
• $\frac{dy}{dx} = \frac{1}{3}x^{-2/3}$

Substituting $\frac{dy}{dx}$ into the formula:

$L = \int_0^1 \sqrt{1 + \left(\frac{1}{3}x^{-2/3}\right)^2} \, dx$

$L = \int_0^1 \sqrt{1 + \frac{1}{9}x^{-4/3}} \, dx$

$L = \int_0^1 \sqrt{\frac{9 + x^{-4/3}}{9}} \, dx$

$L = \frac{1}{3} \int_0^1 \sqrt{9 + x^{-4/3}} \, dx$

This integral does not have a closed-form solution in terms of elementary functions, so we evaluate it numerically.

Numerical Evaluation

We calculate:

$L = \frac{1}{3} \int_0^1 \sqrt{9 + x^{-4/3}} \, dx$from scipy.integrate import quad import numpy as np

Define the integrand

def arc_length_integrand(x): return np.sqrt(9 + x**(-4/3))

Compute the integral numerically

length, _ = quad(arc_length_integrand, 0, 1)

Multiply by 1/3 as per the formula

length *= 1/3 length1.54786565468359781.5478656546835978successThe length of the curve $y = x^{1/3}$ from $(0, 1)$ to $(1, 0)$ is approximately $1.548$.

Would you like to explore the derivation further or see the calculation steps in more detail?

Here are some related questions for further understanding:

1. How is the arc length formula derived?
2. What are some applications of arc length in physics or engineering?
3. Can you calculate the length of a different curve, like $y = \sqrt{x}$?
4. How does numerical integration work for evaluating arc lengths?
5. Could the same curve be evaluated parametrically instead?

Tip: If the integrand is too complex, numerical methods like Simpson's rule or Monte Carlo integration can provide efficient solutions.