To solve this problem, we need to find the length of the curve defined parametrically by the astroid equations:

$x = a \cos^3 t, \quad y = a \sin^3 t, \quad 0 \leq t \leq 2\pi$

Step-by-Step Solution

1. Calculate $\frac{dx}{dt}$ and $\frac{dy}{dt}$:

   Given: $x = a \cos^3 t \quad \Rightarrow \quad \frac{dx}{dt} = 3a \cos^2 t (-\sin t) = -3a \cos^2 t \sin t$

   And: $y = a \sin^3 t \quad \Rightarrow \quad \frac{dy}{dt} = 3a \sin^2 t \cos t$

2. Compute $\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2$:

   We need to evaluate: $\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2$

   Substituting the derivatives: $\left( -3a \cos^2 t \sin t \right)^2 + \left( 3a \sin^2 t \cos t \right)^2$

   Expanding each term: $= 9a^2 \cos^4 t \sin^2 t + 9a^2 \sin^4 t \cos^2 t$

   Factor out $9a^2 \cos^2 t \sin^2 t$: $= 9a^2 \cos^2 t \sin^2 t (\cos^2 t + \sin^2 t)$

   Since $\cos^2 t + \sin^2 t = 1$: $= 9a^2 \cos^2 t \sin^2 t$

3. Set Up the Integral for Arc Length:

   The arc length $s$ of a parametric curve from $t = 0$ to $t = 2\pi$ is given by: $s = \int_0^{2\pi} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

   Substitute $\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 = 9a^2 \cos^2 t \sin^2 t$: $s = \int_0^{2\pi} \sqrt{9a^2 \cos^2 t \sin^2 t} \, dt$

   Simplify the square root: $s = \int_0^{2\pi} 3a |\cos t \sin t| \, dt$

   Since $|\cos t \sin t| = \frac{1}{2} |\sin 2t|$, we get: $s = \int_0^{2\pi} \frac{3a}{2} |\sin 2t| \, dt$

4. Evaluate the Integral:

   By symmetry and periodicity, the integral can be evaluated over one period $[0, 2\pi]$, leading to a result of $6a$.

Final Answer

The length of the circumference of the astroid is: $\boxed{6a}$

Would you like further details on any step, or have any questions?

Here are five related questions for further exploration:

1. How would the arc length formula change if the parametric equations were different?
2. What is the role of symmetry in simplifying integrals in parametric curves?
3. How would we compute the area enclosed by the astroid?
4. Can the arc length integral be evaluated without breaking it down into trigonometric identities?
5. How does the arc length calculation differ for curves defined implicitly?

Tip: Always check for symmetry in the parametric equations when calculating integrals, as it can simplify the problem significantly.