872. Compute the line integral \( I=\int_{L}(x y-1) d x + x^{2} y d y \) from point \( A(1 ; 0) \) to point \( B(0 ; 2) \):
1) along the line \( 2 x+y=2 \);
2) along the arc of the parabola \( 4 x+y^{2}=4 \);
3) along the arc of the ellipse \( x=\cos t ; y=2 \sin t \) (Fig. 185).
Solution. 1) Using the given equation of the line of integration, we transform the curvilinear integral into an ordinary definite integral with variable $x$, then compute it:

$$
y=2-2 x, \quad d y=-2 d x
$$

$$
\begin{aligned}
& i_{1}=\int_{x_{A}}^{x_{R}}[x(2-2 x)-1] d x+x^{2}(2-2 x)(-2 d x)= \\
= & \int_{1}^{0}\left(4 x^{3}-6 x^{2}+2 x-1\right) d x=x^{4}-2 x^{3}+x^{2}-\left.x\right|_{1} ^{0}=1
\end{aligned}
$$

2) Here it is convenient to transform the curvilinear integral into an ordinary integral with variable $y$:

$$
\begin{aligned}
& x=1-\frac{y^{2}}{4}, d x=-\frac{y}{2} d y \\
& I_{2}=\int_{y_{A}=0}^{y_{B}=2}\left[\left(1-\frac{y^{2}}{4}\right) y-1\right]\left(-\frac{y}{2} d y\right)+ \\
& +\left(1-\frac{y^{2}}{4}\right)^{2} y d y=\int_{0}^{2}\left(\frac{y^{5}}{16}+\frac{y^{4}}{8}-\frac{y^{3}}{2}-\frac{y^{2}}{2}+\frac{3 y}{2}\right) d y= \\
& =\frac{y^{6}}{96}+\frac{y^{5}}{40}-\frac{y^{4}}{8}-\frac{y^{3}}{6}+\left.\frac{3 y^{2}}{4}\right|^{2}=-\frac{1}{5}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_8a069bcd8684ea7ea3dbg-302.jpg?height=347&width=227&top_left_y=854&top_left_x=959)

Fig. 185

3) Transform the given integral into an ordinary one with variable $t$, then compute it: $x=\cos t, d x=-\sin t d t ; y=2 \sin t$, $d y=2 \cos t d t:$

$$
\begin{aligned}
& I_{3}=\int_{t_{A}=0}^{t_{B}=\frac{\pi}{2}}(\cos t \cdot 2 \sin t-1)(-\sin t d t)+\cos ^{2} t \cdot 2 \sin t \cdot 2 \cos t d t= \\
& =\int_{0}^{\frac{\pi}{2}}\left(4 \cos ^{3} t \sin t+\sin t-2 \sin ^{2} t \cos t\right) d t=-4 \int \cos ^{3} t d \cos t+ \\
& +\int \sin t d t-\left.2 \int \sin ^{2} t d \sin t\right|_{0} ^{\frac{\pi}{2}}=-\cos ^{4} t-\cos t-\left.\frac{2}{3} \sin ^{3} t\right|_{0} ^{\frac{\pi}{2}}=\frac{4}{3} .
\end{aligned}
$$

(The values of the parameter $t$ at points $A$ and $B$ are found from the given parametric equations of the ellipse using the known coordinates of these points.)