Example. Calculate the length of the arc of the curve given by the equation in polar coordinates

$$
\varrho=6 \sin \varphi, \quad 0 \leq \varphi \leq \pi / 3
$$
Solution.

1. Find $\varrho^{\prime}(\varphi)$:

$$
\varrho^{\prime}(\varphi)=6 \cos \varphi
$$

2. Calculate the differential of arc length:

$$
d l=\sqrt{\varrho(\varphi)^{2}+\varrho^{\prime}(\varphi)^{2}} d \varphi=\sqrt{36 \sin ^{2} \varphi+36 \cos ^{2} \varphi} d \varphi=6 d \varphi
$$

3. Find the arc length by evaluating the definite integral (1):

$$
l=\int_{0}^{\pi / 3} 6 d \varphi=\left.6 \varphi\right|_{0} ^{\pi / 3}=2 \pi
$$

Answer. $l=2 \pi$ units of length.

Conditions of the Problems. Calculate the lengths of the arcs of the given curves.

1. $\varrho=\varphi$
$0 \leq \varphi \leq 1$.
2. $\varrho=\cos \varphi$,
$0 \leq \varphi \leq \frac{\pi}{2}$.
3. $\varrho=2 \sin \varphi$,
$\frac{\pi}{2} \leq \varphi \leq \pi$.
4. $\varrho=1-\cos \varphi, \quad 0 \leq \varphi \leq \pi$.
5. $\varrho=1+\cos \varphi, \quad 0 \leq \varphi \leq \pi$.
6. $\varrho=e^{2 \varphi}$,
$0 \leq \varphi \leq 2 \pi$.
7. $\varrho=1+\sin \varphi,-\frac{\pi}{2} \leq \varphi \leq \frac{\pi}{2}$.
8. $\varrho=1-\sin \varphi, \quad \frac{\pi}{2} \leq \varphi \leq \frac{3 \pi}{2}$.
9. $\varrho=3 \sin \varphi, \quad 0 \leq \varphi \leq \frac{\pi}{6}$.
10. $\varrho=3 \cos \varphi$
$0 \leq \varphi \leq \frac{\pi}{3}$.

Answers. 1. $[\sqrt{2}+\ln (1+\sqrt{2})] / 2 . \quad$ 2. $\pi / 2 . \quad 3 . \pi .4 .2 . \quad 5.2$. 6. $\sqrt{5}\left(e^{4 \pi}-1\right) / 2$. 7. 2. 8. 2. 9. $\pi / 2$. 10 . $\pi$.

### 8.11. Calculation of Volumes by Cross-Sectional Areas

Problem Statement. Calculate the volume of a body if the areas of its cross-sections are known.

Plan of Solution. If $S=S(x)$ is the area of the cross-section of the body by a plane perpendicular to the $O X$ axis and intersecting it at a point with abscissa $x$, then the volume of the part of the body enclosed between the planes $x=x_{1}$ and $x=x_{2}$ is determined by the formula

$$
V=\int_{x_{1}}^{x_{2}} S(x) d x
$$

1. Find $S(x)$.
2. Substitute $S(x)$ into formula (1) and evaluate the definite integral.

Write the answer, including the units of measurement.

Note. The problem is solved similarly if the areas of the sections by planes perpendicular to the $O Y(S(y))$ or $O Z(S(z))$ axes are known.