29.37. Calculate the length of the astroid given by the equation $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$.
29.37. Ans: $6 a$. The astroid can be given parametrically: $x=a \sin ^{3} t, y=a \cos ^{3} t$. It is sufficient to compute the length of a quarter of the astroid, for which $x$ and $y$ are non-negative. Clearly, $x'=$ $=3 a \cos t \sin ^{2} t$ and $y'=-3 a \sin t \cos ^{2} t$. Therefore, $\sqrt{x'^{2}+y'^{2}}=3 a \sin t \cos t$, so the length of a quarter of the astroid is

$$
3 a \int_{0}^{\pi / 2} \sin t \cos t d t=\left.\frac{3 a}{2} \sin ^{2} t\right|_{0} ^{\pi / 2}=\frac{3 a}{2} .
$$

The length of the entire astroid is $6 a$.