Example 12. The arc of the cubic parabola $y=\frac{1}{3} x^{3}$, enclosed between the points $O(0,0)$ and $A(1,1 / 3)$, rotates around the $O x$ axis. Find the area of the surface of revolution.
Solution. According to formula (9), we get:

$$
\begin{aligned}
& \qquad S=2 \pi \int_{0}^{1} \frac{1}{3} x^{3} \sqrt{1+x^{4}} d x=\left.\frac{\pi}{9}\left(1+x^{4}\right)^{3 / 2}\right|_{0} ^{1}=\frac{\pi}{9}(2 \sqrt{2}-1) \\
& \text { Answer. } S=\frac{\pi}{9}(2 \sqrt{2}-1)
\end{aligned}
$$

## Exercises

(below, all parameter values are considered positive)

Calculate the areas of figures bounded by the given curves (1-5).

1. $y=2 x^{2}+x+1, \quad y=0, \quad x=1, \quad x=3$.
2. $y=\frac{1}{2} x^{2}+1, y=x+3$.
3. $y=\sin x, y=\cos x, x=0, x=2 \pi$.
4. $y=\ln (x+6), y=3 \ln x, \quad y=0, \quad x=0$.
5. $y=6 x^{2}-5 x+1, y=\cos \pi x, x=0, x=1$.
6. Find the area of the figure bounded by the ellipse with parametric equations $x=a \cos t, y=b \sin t$.
7. Find the area of the figure bounded by the Bernoulli lemniscate $r^{2}=a^{2} \cos 2 t$.
8. Find the area of the figure bounded by the cardioid

$$
\left\{\begin{array}{l}
x=a(2 \cos t-\cos 2 t) \\
y=a(2 \sin t-\sin 2 t)
\end{array}\right.
$$

9. Find the area of the figure bounded by the curve $r=a \sin 3 t$.
10. Find the area of the figure bounded by the Pascal snail $r=2+$ $+\cos