Example 4. Calculate the area of the figure bounded by the astroid $x=4 \cos ^{3} t, y=4 \sin ^{3} t$.
Solution. Due to the symmetry of the astroid relative to the coordinate axes (Fig. 2.8), it is sufficient to compute one quarter of the area and then multiply the result by 4.

We use the formula $S=4 \int_{0}^{4} y d x$ (see

![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-038.jpg?height=444&width=445&top_left_y=442&top_left_x=78)

Fig. 2.8

$\left.+\frac{1}{2}(\cos 6 t+\cos 2 t)\right] d t=\left.12\left(t-\frac{\sin 4 t}{4}-\frac{\sin 2 t}{4}+\frac{\sin 6 t}{12}\right)\right|_{0} ^{\pi / 2}=6 \pi$

Answer. $S=6 \pi$.