$$ $$

$ \text { The curve is Astroid } \\ $

$ \text {The Parametric equations are } x = acos^3\theta, y = bsin^3\theta \\ $

$ S = 4 \int_0^{\frac{\pi}{2} } \sqrt {(\frac{dx}{d\theta})^2 + (\frac{dy}{d\theta})^2 } \\ $

$ S = 4 \int_0^{\frac{\pi}{2} } \sqrt {3acos^2\theta sin\theta)^2 + (3asin^2\theta sin\theta)^2 } d\theta \\ $

$ = 4 * 3a \int_0^{\frac{\pi}{2}} sin\theta cos\theta d\theta \\ $

$ = 6a \int_0^\frac{\pi}{2} sin2\theta d\theta \\ $

$ = 6a [-\frac{cos\theta}{2}]_0^\frac{\pi}{2} \\ $

$ = 3a [ -cos\pi + cos0 ] = 3a[1+1] = 6a \\ $