Example 2. Calculate the line integral

$$
\int_{L}(x-y) d x+d y+z d z
$$

from point $M(2,0,4)$ to point $N(-2,0,4)(y \geq 0)$ along the curve $L$, formed by the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $z=4$,
Solution. In the section, we obtain a circle

\[
\left\{\begin{array}{l}
x^{2}+y^{2}=4 \\
z=4
\end{array}\right.
\]

Therefore, the parametric equations of the curve \( L \) are

\[
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=4
\end{array}\right.
\]

1. We compute: \( x^{\prime}(t)=-2 \sin t \), \( y^{\prime}(t)=2 \cos t \), and \( z^{\prime}(t)=0 \).

We determine \( t_{1} \) and \( t_{2} \) from the conditions

\[
\left\{\begin{array} { l } 
{ 2 = 2 \operatorname { c o s } t _ { 1 } , } \\
{ 0 = 2 \operatorname { s i n } t _ { 1 } , } \\
{ 4 = 4 ; }
\end{array} \quad \left\{\begin{array}{l}
-2=2 \cos t_{2} \\
0=2 \sin t_{2} \\
4=4
\end{array}\right.\right.
\]

Considering that \( y \geq 0 \), we get \( t_{1}=0 \) and \( t_{2}=\pi \).

2. We compute the line integral using formula (1):

\[
\int_{L}(x-y) d x+d y+z d z=\int_{0}^{\pi}[(2 \cos t-2 \sin t)(-2 \sin t)+2 \cos t] d t=2 \pi
\]

Answer. \(\int_{L}(x-y) d x+d y+z d z=2 \pi\).

Conditions of the Problem. Compute the line integrals.

1. \(\int_{L}\left(y^{2}-z^{2}\right) d x+2 y z d y-x^{2} d z, L\) — part of the curve \(x=t, y=t^{2}\), \(z=t^{3}\) from point \(M(0,0,0)\) to point \(N(1,1,1)\).
2. \(\int_{L}(2-y) d x+x d y, \quad L-\) arc of the cycloid \(\quad x=t-\sin t\), \(y=1-\cos t \quad(0 \leq t \leq 2 \pi)\).
3. \(\int_{L} y d x+z d y+x d z, \quad L-\) first turn of the helical line \(x=\cos t, y=\sin t, z=t\).
4. \(\int_{L}(x+y) d x+(x-y) d y, L\) - circle \((x-1)^{2}+(y-1)^{2}=4\).
5. \(\int_{L} \frac{(x+y) d x-(x-y) d y}{x^{2}+y^{2}}, L-\) circle \(x^{2}+y^{2}=1\).
6. \(\int_{L}(4 x+y) d x+(x+4 y) d y, \quad L-\) part of the parabola \(y=x^{4}\) from point \(M(1,1)\) to point \(N(-1,1)\).
7. \(\int_{L} 2 x y d x+x^{2} d y, \quad L-\) segment of the straight line from point \(M(0,0)\) to point \(N(1,1)\).
8. \(\int_{L} y d x+x d y+(x+y+z) d z, \quad L-\) segment of the straight line from point \(M(2,3,4)\) to point \(N(3,4,5)\).
9. \(\int_{L} z d x+x d y+y d z, L-\) line of intersection of the cylinder \(x^{2}+y^{2}=4\) and the plane \(z=0\).
10. \(\int_{L} y d x-x d y+z d z, L-\) line of intersection of the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cone \(x^{2}+y^{2}=z^{2} \quad(z \geq 0)\).

Answers. 1. \(1 / 35\). 2. \(-2 \pi\). 3. \(-\pi\). 4. 0. 5. \(-2 \pi\). 6. \(-2\). 7. 1. 8. \(33 / 2\). 9. \(4 \pi\). 10. \(-4 \pi\).

## Chapter 10  SERIES

In studying the topic of SERIES, you will become familiar with the concepts of convergence, sum, and remainder of a series, learn how to investigate the convergence of numerical series using comparison theorems and various convergence tests. You will become familiar with the concepts of functional and power series, learn how to find their intervals of convergence and sums. You will learn what a Taylor series is, how to find the Taylor series expansions of functions, and how to apply these expansions in approximate calculations.

Using the package RESHEBNIK.VM, you can compute limits, perform approximate calculations, and verify the results you have obtained.

## 10.1. The Concept of the Sum of a Series

Problem Statement. Find the sum of the series

\[
\sum_{n=1}^{\infty} \frac{A}{n^{2}+p n+q}
\]

where \(A, p, q\) are integers.

Plan of Solution. The sum of the series \(\sum_{n=1}^{\infty} a_{n}\) is the limit \(S\) of the sequence of its partial sums \(\left\{S_{n}\right\}\), i.e.,

\[
S=\lim _{n \rightarrow \infty} S_{n}
\]

where \(S_{n}=a_{1}+a_{2}+\ldots+a_{n}\).

1. According to the problem statement,

\[
a_{n}=\frac{A}{n^{2}+p n+q}
\]

If the roots of the denominator differ by an integer, i.e., \(n^{2}+p n+q=(n+a)(n+a+k)\), where \(k\) is a natural number, then the terms of the sequence of partial sums of the series \(\sum_{n=1}^{\infty} a_{n}\) can be easily found, as many terms in the expression \(S_{n}=a_{1}+a_{2}+\ldots+a_{n}\) cancel each other out.

2. Decompose the general term of the series into elementary fractions:

\[
\frac{A}{n^{2}+p n+q}=\frac{1}{k}\left(\frac{A}{n+a}-\frac{A}{n+a+k}\right)
\]

and write out several terms of the series so that it is clear which terms cancel out when computing the partial sums of the series.

3. Find the \(n\)-th partial sum of the series:

\[
S_{n}=a_{1}+a_{2}+\ldots+a_{n}
\]

by canceling the corresponding terms.

4. Compute the sum of the series using formula (1)

\[
S=\lim _{n \rightarrow \infty} S_{n}
\]

and write the answer.