Example 4. Calculate the area of the figure bounded by the curve $y^{2}=x^{3}$, the line $y=1$, and the $O y$ axis (Fig. 5.8).
Solution. We will determine the required area using formula (5.16). Substituting the values $c=0, d=1$ and the expression $x=y^{\frac{2}{3}}$, we get

$$
S=\int_{0}^{1} y^{\frac{2}{3}} d y=\left.\frac{y^{\frac{5}{3}}}{\frac{5}{3}}\right|_{0}^{1}=\left.\frac{3}{5} y^{\frac{5}{3}}\right|_{0}^{1}=\frac{3}{5}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-210.jpg?height=421&width=489&top_left_y=1281&top_left_x=28)

Fig. 5.9