Example 3. Calculate the line integral

$$
\int_{L} \operatorname{arctg} \frac{y}{x} d l
$$

where $L$ is the part of the Archimedean spiral $\varrho=\varphi(0 \leq \varphi \leq \pi / 2)$.
Solution.

1. We calculate: $\varrho^{\prime}(\varphi)=1, d l=\sqrt{\varphi^{2}+1} d \varphi$ and $f(x, y)=\varphi$, since $\operatorname{arctg}(\operatorname{tg} \varphi)=\varphi$ for $0 \leq \varphi \leq \pi / 2$.
2. Substitute these results into formula ( $1^{\prime \prime}$ ) and compute the definite integral:

$$
\int_{L} \operatorname{arctg} \frac{y}{x} d l=\int_{0}^{\pi / 2} \varphi \sqrt{\varphi^{2}+1} d \varphi=\left.\frac{1}{3}\left(\varphi^{2}+1\right)^{3 / 2}\right|_{0} ^{\pi / 2}=\frac{\left(\pi^{2}+4\right)^{3 / 2}-8}{24}
$$

Answer. $\int_{L} \operatorname{arctg} \frac{y}{x} d l=\frac{\left(\pi^{2}+4\right)^{3 / 2}-8}{24}$.

Conditions of the Problems. Compute the line integrals.

1. $\int y^{2} d l, L-$ part of the curve $x=t-\sin t, y=1-\cos t$ $(0 \leq t \leq 2 \pi) \quad($ arc of the cycloid).
2. $\int_{L} z d l, \quad L-$ part of the curve $x=t \cos t, \quad y=t \sin t, \quad z=t$ $(0 \leq t \leq \pi)$ (first turn of the conical helical line).
3. $\int_{L}\left(x^{2}+y^{2}+z^{2}\right) d l, \quad L-$ part of the curve $x=\cos t, \quad y=\sin t$, $z=2 t(0 \leq t \leq \pi)$ (first turn of the helical line).
4. $\int_{L} \sqrt{2 y} d l, \quad L-$ part of the curve $x=t, \quad y=t^{2} / 2, \quad z=t^{3} / 3$ $(0 \leq t \leq 1)$.
5. $\int_{L} \sqrt{y} d l, \quad L-$ part of the parabola $y=x^{2}$ from point $A(0,0)$ to point $B(2,4)$.
6. $\int_{L} x d l, \quad L-$ segment of the line from point $A(1,0)$ to point $B(0,2)$.
7. $\int_{L}(x+y) d l, \quad L-$ boundary of the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$.
8. $\int_{L} \sqrt{x^{2}+y^{2}} d l, \quad L-$ circle $x^{2}+y^{2}=2 x$.
9. $\int_{L} \sqrt{x^{2}+y^{2}} d l, \quad L-$ cardioid $\varrho=1-\cos \varphi$.
10. $\int_{L}|y| d l, \quad L-$ Bernoulli's lemniscate $\varrho=\sqrt{\cos 2 \varphi}$.

Answers. 1. $\frac{64}{3}$. 2. $\frac{\left(2+\pi^{2}\right)^{3 / 2}-2 \sqrt{2}}{3}$. 3. $\frac{2 \pi \sqrt{5}\left(3+16 \pi^{2}\right)}{3}$.

4. $\frac{1}{8}\left(3 \sqrt{3}-1+\frac{3}{2} \ln \frac{3+2 \sqrt{3}}{3}\right) \cdot 5 \cdot \frac{1}{12}\left(17^{3 / 2}-1\right) \cdot 6 \cdot \frac{\sqrt{5}}{2} \cdot 7 \cdot 1+\sqrt{2}$.
5. 8. 9. $\frac{32}{3}$. 10. $2(2-\sqrt{2})$.

## 9.2. Line Integrals of the Second Kind

PROBLEM STATEMENT. Compute the line integral

$$
\int_{L} P(x, y, z) d x+Q(x, y, z) d y+R(x, y, z) d z
$$

where $L-$ is a part of a smooth curve defined parametrically

$$
\left\{\begin{array}{l}
x=x(t) \\
y=y(t), \\
z=z(t)
\end{array} \quad t_{1} \leq t \leq t_{2},\right.
$$

PLAN OF SOLUTION. The line integral of the second kind over the curve $L$ is defined by the formula

$$
\begin{aligned}
& \int_{L^{L}} P(x, y, z) d x+Q(x, y, z) d y+R(x, y, z) d z= \\
& \int_{t_{1}}^{t_{2}}\left[P(x(t), y(t), z(t)) x^{\prime}(t)+Q(x(t), y(t), z(t)) y^{\prime}(t)+R(x(t), y(t), z(t)) z^{\prime}(t)\right] d t .
\end{aligned}
$$

1. Compute $x^{\prime}(t), y^{\prime}(t)$ and $z^{\prime}(t)$.
2. Compute the line integral using formula (1) and write the answer.

REMARK 1. If the boundary points of the curve $L M\left(x_{1}, y_{1}, z_{1}\right)$ and $N\left(x_{2}, y_{2}, z_{2}\right)$ are given in Cartesian coordinates, then $t_{1}$ and $t_{2}$ are determined by solving the systems of equations

$$
\left\{\begin{array} { l } 
{ x _ { 1 } = x ( t _ { 1 } ) , } \\
{ y _ { 1 } = y ( t _ { 1 } ) , } \\
{ z _ { 1 } = z ( t _ { 1 } ) ; }
\end{array} \quad \left\{\begin{array}{l}
x_{2}=x\left(t_{2}\right) \\
y_{2}=y\left(t_{2}\right) \\
z_{2}=z\left(t_{2}\right)
\end{array}\right.\right.
$$

REMARK 2. If the curve is given as the intersection of two surfaces:

$$
\left\{\begin{array}{l}
F_{1}(x, y, z)=0 \\
F_{2}(x, y, z)=0
\end{array}\right.
$$

then it must be parameterized.