EXAMPLE 1. Calculate the line integral

$$
\int_{L} \frac{y}{3} d x-3 x d y+x d z
$$

along the part of the curve $L$ given parametrically by

$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t-2 \sin t
\end{array} \quad 0 \leq t \leq \frac{\pi}{2}\right.
$$
## Solution.

1. Compute: $x^{\prime}(t)=-2 \sin t, y^{\prime}(t)=2 \cos t$ and $z^{\prime}(t)=2 \sin t-2 \cos t$.
2. Compute the line integral using formula (1):

$$
\begin{aligned}
& \int_{L} \frac{y}{3} d x-3 x d y+x d z= \\
& =\int_{0}^{\pi / 2}\left[\frac{2}{3} \sin t(-2 \sin t)-6 \cos t(2 \cos t)+2 \cos t(2 \sin t-2 \cos t)\right] d t= \\
& =2-\frac{13}{3} \pi
\end{aligned}
$$

Answer. $\int_{L} y / 3 d x-3 x d y+x d z=2-\frac{13}{3} \pi$.