Example 6. Calculate the length of the curve $r=a \sin ^{3} \frac{\varphi}{3}$.
Solution. The entire curve is described by a point as $\varphi$ changes from 0 to $3 \pi$ (Fig. 5.16).

Fig. 5.16

$$
\text { Since } r^{\prime}=a \sin ^{2} \frac{\varphi}{3} \cos \frac{\varphi}{3} \text {, then based on formula (5.20) we find }
$$

$$
\begin{aligned}
& l=\int_{0}^{3 \pi} \sqrt{a^{2} \sin ^{6} \frac{\varphi}{3}+a^{2} \sin ^{4} \frac{\varphi}{3} \cos ^{2} \frac{\varphi}{3}} d \varphi=a \int_{0}^{3 \pi} \sin ^{2} \frac{\varphi}{3} d \varphi= \\
& =a \int_{0}^{3 \pi} \frac{1-\cos \frac{2 \varphi}{3}}{2} d \varphi=\frac{a}{2} \int_{0}^{3 \pi} d \varphi-\frac{a}{2} \int_{0}^{3 \pi} \cos \frac{2 \varphi}{3} d \varphi= \\
& \quad=\left.\frac{a}{2} \varphi\right|_{0} ^{3 \pi}-\left.\frac{a}{2} \cdot \frac{3}{2} \sin \frac{2 \varphi}{3}\right|_{0} ^{3 \pi}=\frac{a}{2} 3 \pi, l=\frac{3 \pi a}{2}
\end{aligned}
$$

## Problems

Find the length of the arc of the curve:

1. $y=\frac{x^{2}}{2}-\frac{3}{2}$, cut off by the $O x$ axis.
2. $y=2 \sqrt{x}$ from $x=0$ to $x=1$.
3. $y=a \operatorname{ch} \frac{x}{a}$ between the lines $x= \pm a$.
4. $x=\frac{2}{3} y^{\frac{3}{2}}$ from $y=0$ to $y=3$.
5. $x=e^{t} \sin t, y=e^{t} \cos t$ from $t=0$ to $t=\frac{\pi}{2}$.

Calculate the length of the entire curve:
6. $x=a \cos t, y=a \sin t$.
7. $x=a(2 \cos t-\cos 2 t), y=a(2 \sin t-\sin 2 t)$.
8. $x=\frac{c^{2}}{a} \cos ^{3} t, y=\frac{c^{2}}{b} \sin ^{3} t\left(c^{2}=a^{2}-b^{2}\right)$.
9. $r=a(1+\cos \varphi)$.

10. Find the length of the loop of the curve: $x=t^{2}, y=t-\frac{1}{3} t^{3}$.
11. Calculate the length of the arc of the hyperbolic spiral $r \varphi=1$ from point $A\left(2, \frac{1}{2}\right)$ to $B\left(\frac{1}{2}, 2\right)$.
12. Find the length of the arc of the logarithmic spiral $r=a e^{\varphi}$ (Fig. 5.17) lying inside the circle of radius $r=a$.

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-218.jpg?height=310&width=460&top_left_y=1231&top_left_x=42)

Fig. 5.17 Logarithmic

$$
r=a e^{\text {spiral }}
$$

## Answers

1. $2 \sqrt{3}+\ln (2+\sqrt{3})$. Hint.

$\int \sqrt{1+x^{2}} d x$ is calculated by integration by parts (see Example 6, § 9.4). 2. $\sqrt{2}+\ln (1+\sqrt{2})$. Hint. When calculating the integral $\int_{0}^{1} \sqrt{\frac{1+x}{x}}$, apply the substitution $\quad x=\operatorname{sh}^{2} t$.

3. $2 a \operatorname{sh} 1 \approx 2.35 a$. 4. $9 \frac{1}{3}$. Hint. The length of the arc of the curve $x=\varphi(y)$ is calculated using the formula $l=\int_{c}^{d} \sqrt{1+x^{\prime 2}} d y .5 . \sqrt{2}\left(e^{\frac{\pi}{2}}-1\right)$. The curve is called a logarithmic spiral (Fig. 5.17), its equation in polar

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-219.jpg?height=93&width=1085&top_left_y=317&top_left_x=29)
Note. The limits of integration are determined from the equation $t-\frac{1}{3} t^{3}=0$. 12. $a \sqrt{2}$.

## § 5.4. Volume of a Solid of Revolution

The volume of the solid obtained by rotating the curvilinear trapezoid $A a b B$ (Fig. 5.18), where $A B$ is the arc of the curve $y=f(x)$, around the $O x$ axis is calculated using the formula

$$
V_{x}=\pi \int_{a}^{b} y^{2} d x \text { or } V_{x}=\pi \int_{a}^{b} f^{2}(x) d x
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-219.jpg?height=422&width=491&top_left_y=1005&top_left_x=39)

Fig. 5.18

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-219.jpg?height=414&width=492&top_left_y=1002&top_left_x=612)

Fig. 5.19

The volume of the solid obtained by rotating the curvilinear trapezoid $C c d D$ (Fig. 5.19), where $C D$ is the arc of the curve $x=\varphi(y)$, around the $O y$ axis is determined by the formula

$$
V_{y}=\pi \int_{c}^{d} x^{2} d y \text { or } V_{y}=\pi \int_{c}^{d} \varphi^{2}(y) d y
$$